Reddit mentions of Working Construction: Why White Working-Class Men Put Themselves―and the Labor Movement―in Harm's Way

Political Scientist here. I'm not an expert on the minutae of either movement, but I feel that I have a pretty solid grasp of the broad strokes. My own research examines the role of gender in white supremacist communities.

I don't know that each 'side' has an obligation to make their ideas more palatable, but I do feel that each needs to ensure that their claims are backed up by serious quantitative (and qualitative, to an extent) evidence.

In the case of feminism, the concept - and later movement - arose from the recognition that there are definite, demonstrable and observable inequalities in society (specifically European and North American) which placed women in the position of 'inferior' citizens - if they were even considered citizens at all. Women could not vote or hold elected office; it was exceedingly rare for a woman to run a company or to even be educated, since many colleges and universities were 'men only'. These weren't hypothetical problems, they were tangible and endemic. It is a simple fact that for much of the history of western society, women have been considered to be of less 'worth' than men. This attitude has changed to an extent, but it is still the case that women are often at a disadvantage in many sectors of society - from being encouraged to enter 'feminized' careers like secretaries, nurses, teachers, etc, to often being dissuaded from entering other sectors of the workforce - IT, Industry, the Military, etc. While on paper, women have been afforded the same legal rights as men, in reality there are still observable structural barriers for women that do not exist for men.

The "Men's Rights" movement, on the other hand, is an ideology based on the assumption that as women have gained rights in society, men have been 'losing' theirs. The most commonly cited examples of this 'discrimination' are, as you've pointed out, so-called 'false rape' accusations and parental rights. More often than not, allegations of 'false rape' are based on anecdotal evidence and are therefore untrustworthy as a way of determining actual statistics on the phenomenon - although I will provide a source below which seems to indicate that such 'false rape' accusations account for less than 8% of all accusations.

While in some respects the movement has attempted to 'piggyback' itself off of more established rights movements (such as the civil rights, gay rights and women's rights), it nevertheless suffers from a crisis of evidence. Put simply, there is currently zero evidence to suggest that men are in any way the victims of widespread, systemic discrimination based on their gender. While there are instances of men receiving short shrift at the hands of judges who perhaps feel that women are simply 'better' or more 'naturally' able to care for children, these do not prove the existence of systemic or structural discrimination. (Look at Myths 4, 5, 6, specifically)

As some scholars and authors have noted (see below), the crises of masculinity so often noted and lamented by Men's Rights activists can be better explained by examining the sexist social structures put in place by other men, rather than lamenting the 'feminization' of society. Why does it seem that men lose out in custody battles? Maybe because men have been taught from an early age that women are more 'nurturing' or possess a 'maternal instinct' that makes them natural parents - which is an essentialist position that demands that women behave in a certain way. This same belief can also affect how judges (who are mostly male) choose to assign custody in cases where they are forced to.

Put simply, while feminist critiques of society are based on observable instances of institutionalized sexism, the Men's Rights movement, and its attendant critiques are not.

Additional Sources:

Cancian, Maria, Meyer, Daniel R. “Who Gets Custody?”, Demography, Volume 35-Number 2, May 1998

Philip N.S. Rumney (2006). FALSE ALLEGATIONS OF RAPE. The Cambridge Law Journal, 65 , pp 128-158 doi:10.1017/S0008197306007069

Who Cares? The classed nature of childcare

Kris Paap "Working Construction: Why White Working-Class men put Themselves -- and the Labor Movement -- in Harm's Way"

R.W. Connell "The Men and the Boys"

R.W. Connell "Masculinities"