It's not just about people lying though. I could see plenty of men thinking that they would happily take on the stay at home role, only to be slapped upside the head with the halibut of reality when they see what that actually means. Because we as a society don't value or acknowledge this work, it's probably very easy to think it's folding some laundry, grocery shopping, scrubbing a toilet, and then eating bon bons in all the remaining free time. Since we don't talk about accurate representations of what the life is actually like, and since even the people in the roles often don't fully realize the extent of the work they do ("Oh, I never thought about how I have to pay attention to what junior talks about in the months up to his birthday so I can know what gifts to get him, and then search to find those items, and how that's a time and energy suck." <--this, times 1000), someone could in good faith say they are eager to take on the role, only to find that the actual expectations are intolerable.
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Dropping the traditional roles isn't quite enough though. If women who stay at home or work part time are taking on more than an even share of life's work, it doesn't become much better if suddenly a bunch of men are also taking on more than their fair share. Yes, it would be great it we movd away from the gendered split of responsibilities and expectations. But we also need to start valuing that part of the work, and respecting it, and acknowledging it, and counting it when partners are looking at how to split responsibilites.
Now, I suspect that if men started filling these roles in significant numbers, some of that would happen automatically. But I don't think it becomes a utopia when some men are being saddled with the unfair expectations that women have dealt with for centuries.
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I have a friend who is currently solo parenting. Being surrounded by military families, this isn't especially uncommon. Only in this case, her husband is not on planet earth. I can not imagine the stress of raising young kids without help, without regular communication (though when DH and I started our military journey, email was just becoming available on ships and was far from reliable), and with the layer of extra fear that comes from work that's dangerous even to those used to military experiences. Finding someone willing to sacrifice at that level--phew. But we need those people if we want other people to be able to travel to space, or work on an oil rig, or be in the military. Sure, some single people would do those things, but we need more people than just that pool. And yet we minimize the role of the people--almost always women--who fill those roles.