Your analysis of the problem is excellent but I think here you overdramatize the situation as you move towards solutions.
The problem with adjusting wages is more that we don't have a robust mechanism to do it than that it necessarily causes a problem that other methods don't cause.
The whole point of money is that it's extremely fungible. Whether parents get more than they used to because their wages are higher, or because they get a tax break, or becomes the government just flat out gives them money, or because services that otherwise would have to be paid for are free, it takes only a few minutes of effort for them all to result in the same thing: a higher balance in a checking account somewhere, enabling higher spending.
The question is really: of those mechanisms, which is easiest to balance for parity between people without children and people with children?
Given that we generally let the market set wages, and whenever you don't the market finds very creative ways to normalize the difference, wages are probably out. That leaves tax breaks, grants, and services.
But the tax break absolutely does give the parents more spending power, and if it's too big could push some individuals into poverty. It being a tax break doesn't mean that it doesn't impact the economy! It's just plain more money to spend--not even buffered from the rest of the economy, slightly, by being a service.
Out of the four options, you've leaned into tax breaks the most, but I think when applicable, services are easier to use to generate a "similar lifestyle". If you offer anyone who has a child a free crib, for instance, except for the impact on the crib market (which could be sizable), most of the economy would then be blind to the difference between parents' and non-parents' economic demand for cribs. If daycare was free, and stay-at-home parents could collect the daycare expense personally, the economy would be largely blind to childcare costs (again, except for the childcare market itself, which would be strongly affected).
Again, it's not that the cost of these services can't, by virtue of money, be rendered fungible. The point is that because it is a direct need that arises from having children, simply fulfilling that need automatically balances the economic impact.
At some point, this starts getting difficult to implement. People with children need a larger house. Teenagers eat a LOT. If you go on vacation, staying in a hotel gets more expensive if you won't fit in a standard-sized room. And so on. So the government using tax policy to try to balance things also makes sense rather than trying to intervene in so many different parts of the economy. The United States, for instance, does try to use the tax code, it just doesn't try hard enough: you get only a $2000 tax credit per child, which is less than 1/5th of what people spend in practice. The U.K. is a bit more generous, but still offers drastically less than actual expenses.