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.
Flooding the market with new labor increases the supply Against a fixed demand, this lowers wages. So everyone not getting the subsidy feels pressure from stagnating wages plus the increased tax burden.
Let's assume that all those new laborers get paid and therefore demand also increases, moving the equilibrium so some of the wage stagnation pressure is dampened. It's still not going to offset the effect of new labor and taxes.
All this does is modify the equilibrium of supply and demand in the market such that those not receiving the subsidies (or evem those not receiving as much subsidies as others) are negatively impacted through lifestyle discrimination.