What exactly is the argument? That more religion results in greater distinction between the acceptable roles of men and women? That's plausible enough. But if the implication is that men opt out of religious participation if it does offer increased subjugation of women -- that is a stretch for me.

That is what many in America are asking themselves. And, speaking as a conservative Christian (both theologically and politically conservative, though no doubt the two almost always go together) that represents a sad state of affairs. So if people generally answer your "real question" in the affirmative, what comes next, logically and sequentially in time? They then ask themselves what I will call my "real real question", which is: Do we, do I, still believe in the free exercise of religion as touted in the Bill of Rights? And I think many must be answering that question in the negative if they support helping out parents with private school tuition, but not if their children attend private religious schools. Or if they think it right that a Christian baker should be forced to participate in the celebration of a gay marriage, rather than just refer the couple to an equally excellent baker who has no moral/religious objections to doing the job.