Since it's subsidizing specific behavior and not merely being poor or whatever, people will naturally look at whether they think that behavior ought to be incentivized, or whether the government should stay neutral.
My wife is also a stay at home mom, and I've argued before that an increase in the child tax credit with a phase out for high income (so we might not qualify) makes more sense than a childcare credit/deduction for this reason. Then you're just subsidizing having kids, which seems fine to me (assuming we're subsidizing anything) since that's sort of necessary to sustain society.
- Bidding up the price of housing
- Fewer parents active in overseeing the schools, volunteering to fix up the community, etc.
- Less general slack for parents to help each other out
- Fewer mom friends around during the day, less social life for existing stay-at-home moms
- Peer pressure and implicit societal pressure to work a career
- Parents sending their kids to camps and aftercare, rather than having kids free-range around the neighborhood and play with friends, so fewer playmates for the non-camp/non-daycare kids.
So we've come to a crossroads where something profoundly un-libertarian is viewed by the anti-libertarians as libertarian because it incidentally achieves some of its aims.