It does seem disingenuous, as no one has ever claimed that "unpaid" work doesn't have value. It does have value, and that's why it's always compensated in one way or another.

In fact, arguably it's highly beneficial to families to have one partner that isn't paid on the books for this work, as that means they don't have to pay taxes on that pay.

My parents both worked, but a lot of my friends had stay-at-home mothers, and they seemed to be doing a lot better financially, perhaps partly due to this effect.

I would argue the exact opposite. Taxes substantially favor two smaller incomes over one bigger one because of tax brackets. Two people earning 30k before taxes will get a lot more money after taxes than one person earning 60k. Because of that, adding a second income of $X adds a lot more money than simply getting a pay increase of $X dollars.

Your example feels more like correlation than causation. The families that were well off could afford for one parent to stay home, while poor families required both parents to work to make ends meet.

That might be so if one could choose to work half-time for half pay. This option is usually not available, though, at least in the US. (Certainly I couldn't, particularly once one rolls in medical insurance effects.) There are pretty good reasons why many positions don't offer this option.

As for my example, I've no idea if the impression was correct. But it's certainly the case that stay-at-home moms stayed at home for cultural reasons and/or because they wanted to, at least where I lived. Being well off (ha) enabled this option, but it certainly didn't drive it.

edit: Note also that your scenario doesn't involve taxing unpaid work, which is what I was discussing. Instead you're just shuffling it around, which is a different argument.