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 $3/day payment because we don't take advantage of free lunch. Having free lunch available is unequivocally a good thing, regardless of whether we personally partake.
And no it's not a free lunch. If stay-at-home in a family isn't reimbursed, they are actually worse off, because now they have an additional tax they are paying that they did not have before. So now even more people like you who wanted a parent to stay at home are driven out of it because their family budget comes upon this tax.
Yes. And no. The gov gives the child X$ per week/month/year. The child parents use that money to take care of the child.
Society benefit from children that are well taken care of. Mechanisms to ensure that they are well taken care of are needed. Well funded daycare centers are one of the mechanism. A well funded household with a parent/grand parent/uncle is another one. In both case, an agency is in place to ensure the wellness level of the child.