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.
I'm willing to accept that position, I'm not necessarily for free childcare, only believe that if childcare is to be free it should follow the child. I don't see at all how a mom taking care of a child "needs" the money less than a daycare worker/company taking care of the child. What you're proposing is just yanking the money away from them in a tax, then lording it over them that they have to take the latter if they want the cash back -- trying to track to which caregiver the money goes instead of just providing the resources for the child and let the parents decide what works best for their family.