It's not a crab bucket mentality. Subsidizing one group that competes in the same markets (e.g. only dual income families, who compete with single income families for housing in desirable areas to raise kids) actually increases costs for the unsubsidized group. It doesn't just make them relatively worse off, but absolutely worse off. It shifts the margin of who can afford a single family lifestyle, all else equal.

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.