My wife is a stay-at-home mom. We are lucky that we can afford to do this. Most of our kid's friends have both parents working and they pay for child care. If suddenly they were able to have that childcare paid for, that would be wonderful! It doesn't affect our situation at all. Why would we oppose it? I don't need to have my own "waiver" payment in order for me to be in favor of my neighbor's burden being lifted.
It's like free school lunch. We pack our kid a lunch every day, but some families rely on the school-provided free lunch. It's never even occurred to me that we should get a $3/day payment because we don't take advantage of free lunch. Having free lunch available is unequivocally a good thing, regardless of whether we personally partake.
Even if we consider it as an "efficiency" problem, it is far cheaper for a person to be paid to take care of N children (where N is not too large), rather than have the have the mom, who is probably qualified in some other field, take care of just their children.
> the mom, who is probably qualified in some other field
Parents are plenty qualified to take care of their kids. And their qualifications in some other field doesn’t mean that working that field is better for them or their kids or the country. Having strong family structures and time together is pretty valuable.
If it costs $100/child at a daycare facility, but $200/child for someone to be a stay-at-home parent, and you're asking me, a random taxpayer, to pay for one of those for someone else, from a financial perspective I will likely prefer to pay for the former.
Now, I personally don't get to decide where tax dollars go, but I could easily imagine there are enough people with this preference that it could influence public policy.
Having said that, if it's actually significantly better for a child to have a SAH parent, I might change my tune. (My mother was a SAHM, and I think that was great for us growing up.)
In your system you’ve created a messed up incentive where parents are better off just sending the kid to the daycare and having the mom sit at home and do absolutely nothing.