It is, sort of, through Dependent Care FSA programs which let you contribute money pre-tax and then spend it on child care. For some reason though, the limit is $5000/year which isn't nearly enough to cover any sort of frequent child care. It's also sort of a pain to administer these programs.

There's also a child care tax credit which gives a similar amount of tax savings. So overall you do have some tax benefits to help with daycare costs, but they could be better.

Yes, making as many kid-related things as possible tax-deductible is a sensible way to provide "subsidies" that incentivize more children.