Well, part of the problem with these kinds of generalisations is that variation within each set is greater than variation between the sets (i.e. "men are better off than women on average" =/= "all men are better off than all women"). Another part of the problem is: which variables count, and how are they ranked? Men are more likely to commit suicide than women - is that a "structural oppression", since it's by their own hand? If so, is it more or less important than a much lower risk of being raped (unless in prison)? Usually the idpol-feminist vs MRA discourse turns into one set of statistics vs another set of statistics, along with endless feuding about which gender is to blame for which social problems, or relative frequency of abuse in each direction.
Keeping these limits in mind (it's *average*, there's lots of individual exceptions, not all the averages run the same way, and causality is often contested), there's still good reasons why the feminist view has more traction in academia and welfarist/NGO circles:
1. There's far more men than women in all the most powerful social positions - presidents and prime ministers in the US and worldwide, CEO's, judges, military generals, chiefs of police.
2. Traditionally masculine jobs which still have greater percentages of men in them (usually STEM-related) are paid much better than traditionally feminine jobs with higher proportions of women (nursing, social work, teaching, retail, care work, etc).
3. Men are paid more than women on average. This is contested because it's quite rare for men and women to do the same work on the same contracts, and if they are, it's illegal to pay them differently. So generally, what happens is that women are given similar jobs at lower points in the pay-scale, or the same jobs but part-time or casual.
4. Again the causality is obscure, but women still end up doing most of the "unremunerated reproductive labour" (childcare, family care, emotional support, household chores) even when they're working the same hours as men, leading to extreme time-scarcity among a sizeable percentage of women.
5. Men still tend to have more power in relationships. There's a lot of exceptions, but the average is still that men have more leisure time, spend more of a couple's income on themselves than women do, and get their own way in disagreements more often.
6. Male violence against women is more common than female violence against men (though this might be levelled-out if passive-aggression and emotional abuse are included). This particularly applies to the more serious forms of violence where someone ends up injured or killed.
7. Men are still a lot more visible in the media and culture, and in particular, women are either absent or objectified in media aimed at men, without the reverse happening to anything like the same degree. About a third of films still don't pass the "Bechdel Test". It's very rare for them not to pass the "reverse Bechdel test".
All of which amounts to persistent structural inequality in my view, especially considering that it's continuous with the pre-1970s period when gender inequalities were a lot more explicit.
Men are more at risk of being victims/survivors of violence than women. Men die younger on average, are at greater risk of suicide and certain health risks, are much more likely to be sent to prison. Traditionally men would be the ones to die in war as well, although today, most of the casualties of war are civilians. It's not clear how far this offsets the above "advantages". For one thing, generally these are not things done to men by women - they're done to men by men, or to men by themselves. But some of the things "done to" women are mainly done by women to themselves or to other women as well (everything from "slut shaming" to genital mutilation to anorexia). I think the evidence fits more closely an account "patriarchy hurts men too" rather than "women oppress men", though it's certainly not as simplistic as idpols would like it to be.
In neoliberalism there's another complication. Neoliberalism rewards subordination and punishes assertion, so women's greater subordination is sometimes an advantage. Some employers prefer to hire women *because* they can be paid less, are less assertive, are more prepared to take on part-time or temporary or casual jobs, or work in traditionally feminised jobs. And this leads to situations where, even though "men's jobs" are still much better-paid and have better conditions than "women's jobs", women are doing better because there's a lot more "women's jobs" going (without this advantage raising pay or conditions in "women's jobs"). Also, traditionally masculine jobs (especially but not only in the rich countries) have been hit much harder by deindustrialisation, financialisation, robotics, anti-union policies, offshoring, and other shifts in the neoliberal period than traditionally feminine jobs.
There's another complication - what idpols would call "intersectional" I think - which is that, in many ways, men at the bottom of the pile are in a structurally very different (and arguably, worse-off) position than women at the bottom of the pile (and certainly than women at the middle or top of the pile). Or if they're equally badly off, it's in very different ways. It's a difference between adverse incorporation (inclusion with worsened exploitaiton) and outright exclusion. Women who are worse-off tend to remain within "society" but in very adverse roles - hyperexploited pink-collar or sweatshop workers, dependent wives or daughters, secondary mistresses of wealthy men, sex workers, benefit claimants. In Africa there's less of a "restive youth problem" among women because they remain part of the father's household if they don't marry or have a career, whereas young men are expected to go out and make their fortune. Women are also likely to end up in transactional relationships (which always seem to be economic male-to-female transfers in return for power-unequal relationships), and/or with access to subsistence resources. It's called the "restive youth problem", because in Africa a man is still considered a "youth" until he has a wife, house, children, and farmland or a career - so there's millions of "youths" in their 20s and 30s and even 40s, who are eternal youths so to speak, and can never make the passage to "manhood" (it's happening worldwide, but the African way of framing it is particularly stark). And the "restive youth" demographic are regularly seen as being at the heart of a whole swathe of destabilising (and for anarchists, sometimes inspiring) social problems - militant protests, criminal gangs, armed opposition, the "hired thugs" of political parties, the people who turn out en masse to protest contested elections, the constituency of the Arab Spring ("we are all Mohammed Bouazizi"). In Egypt the pattern is that neoliberalism has harmed women, destroying most suitable jobs (compatible with traditional reproductive labour roles), but the usual response has been a "return to the home" and reinforcement of traditional gender roles, whereas men are more affected by straightforward exclusion. In China, traditional protected male workers in the rustbelt and coastal cities have been replaced by temporary migrant workers, a small majority of whom are women. In contrast with the older workers, these migrants are not legally resident in the cities, have no de jure social rights or de facto legal rights, and are worked to the point of collapse. But the older male workers have simply been fired. In Lisa McKenzie's work on St Ann's (inner-city post-industrial Britain), it's a similar pattern - most of the men are off the radar, making money from crime or grey-economy stuff, whereas women are on benefits and plugged into the various everyday tentacles of the state - generally because the women have caring responsibilities for children. There's all kinds of strange dynamics resulting from this - men providing assistance informally while women tell the benefits agency they don't know who the father is, women accessing health resources on behalf of men, men using women's homes as unofficial residences to reduce the risk of police raids.
This is also the background to why girls now outperform boys at school: middle-class girls and boys get similar results, but boys from the poorest communities tend to give up early on school and head for the underground economy, whereas girls seek qualifications to get into low-end white-collar work.
Idpols would continue to claim that women are more oppressed, because they don't really recognise exclusion or the excluded as a social category. They'd say that men are dropping out because they have the "privilege" to do so, and also that men are more likely to respond in a "macho" or "entitled" way to conditions of precarity (whereas women tend to buckle down and cope). There might be some truth to this, but it's very much a partial picture. People don't enter the criminal economy, with all its risks and instability, because they're "privileged" - even if it's a better fit, in some ways, with macho values; and it's because of this split that young men also bear the brunt of everything from stop-and-search and mass imprisonment to inter-gang violence and murders by the police. If men are more likely to respond forcefully to stigma and marginalisation, this is a sign that women (on average) are more conservative - and not as idpols like to spin it, that conservative positions are actually "more radical". The entire idpol playbook is premised on inclusion, on the assumption that everyone is "inside" the system no matter how marginally or contradictorily (since there isn't really a system, but only an array of "discourses" which we're never outside). But from a sociological point of view, radical exclusion is "lower" in the social hierarchy than adverse inclusion, and a lot of the statistical biases against men (regarding violence, imprisonment, suicide and so on) stem from the position of this group. So probably - if we drew an "intersectional" group chart with rankings of a dozen or so different class-race-gender intersections, socially excluded young men would actually be at the bottom (even though other groups of men would outrank women of the same class and race).
Just as importantly - from a radical perspective - most of the most militant and effective social movements of the present/recent past (whether progressive or not) are drawn primarily from this excluded young male population. Everything from Greek anarchism to Boko Haram to the Thai redshirts to Anonymous, from the Russian far right to the Nepalese Maoists to MEND, to politicised gangs like the Brazilian PCC. So, if there's a strategic constituency for social change, a constituency most likely to revolt and most likely to reject the whole system, it's excluded young men (and not adversely incorporated women). And this particular combination of exclusion and rebellion (not "oppression within social change movements") is why the group tends to be overrepresented in anarchist spaces. And the question of whether this stratum on the whole turn to anarchism or some other autonomous approach, or to right-wing fascist, ultranationalist, fundamentalist or ethnic formations is politically absolutely vital. In Europe and America and to an extent Latin America, the picture is complicated somewhat because of a partial breakdown of traditional gender roles - there are women in the North whose social situation is the same as that of excluded young men, whereas this rarely happens in Africa or the Middle East for example. I've observed in my circle of friends how young women in this position adopt similar ideologies to men in this position, but how those who become increasingly incorporated (whether it's professional/NGO or family/benefits) tend to move towards idpol and liberalism. This also happens to men who take the same path, but there seem to be more men than women who are still radically excluded as they get older. This is all very much neglected, I think because the included, middle-class people who form the organic intellectual leadership of idpol have little interest in talking about the excluded. And if you look at what group primarily is being excluded by safe space policies and the like, it's quite clearly the same people affected by zero tolerance, quality-of-life policing, and Third Way behaviour-policing: radically excluded young men. Which is partly why, at the moment, anarchism is losing these young men to the far-right and to various ethnic, religious and nationalist projects. It's not just this though, because there's also a tendency for anti-poor ideologies - particularly the idea of the economy, or the welfare state, as an exclusionary "club" which owes something only to club members - are a lot more common among the poor and working-class than among the middle-class (apparently driven by populist media and by a sense of threat and fear arising from intensified competition at the bottom). Anarchism is a "big tent" approach and not a "club" approach (even if one's affinities are limited to one's immediate group), because anarchism rejects market "reciprocity" and state repression of "outsiders" - which paradoxically means that anarchism is in the interest of the excluded and the adversely incorporated, but not necessarily close to their spontaneous ideologies.