It is funny to say this in this specific conversation. The exact logic you are using to support rebates for stay at home parents applies to childless people. So why are you drawing the line exactly where you are drawing it and why is that a better place than where this policy is currently drawing it?
What the government should encourage is charitable donations, and when I say that, I mean the mere act of it. There should be no tax incentive for doing so.
Where children are concerned, if anything, perhaps make the sales tax on child-related services zero, and increase sales tax on luxury goods associated with sink or dink households. At least that methodology provides the opportunity to forgo the penalties.
Of course the tricky thing is that not all children produce positive externalities, some have massively negative externalities and a naive subsidy might encourage the wrong kind of reproduction ...
Anyways, if you don't want any subsidies, one policy change is to eliminate general social security and simply have each retiree get the social security money paid only from their own children. Social security is not a savings plan or insurance, what it actually is is a socialized version of the current generation of children paying for their parents retirement. The non-socialized version is just the parents getting money of the kids that they raised themselves, and if you did not put in the work of raising kids, you don't get social security.
The argument you’re making in general is a valid one about subsidies, it’s a weird argument to make regarding children since having children is the only way society survives. Unless your claim is that we’re overpopulated but generally people in developed countries are not reproducing, and a meaningful part of that does appear to be the cost.
So the answer is that this specific subsidy is net beneficial as we want to make it easier for people to have and raise kids, not least of which because it produces better adults when those kids grow up and makes society healthier.