Use Riester and family support in a targeted manner

I would earn just under 6,000 net after birth + with the child benefit taken into account.

So per child there is 250€ child benefit, thus 750€. Your income is therefore just under 5,250€. To get the subsidy with 3 children, the limit is 90,000€ gross p.a. = 7,500€ gross per month. 7,500€ gross per month bring me only 5,046.99€ net according to my gross-net calculator, even with the cheapest health insurance and no church tax. Did you round up by more than 200€ here or where does the discrepancy come from?

Possibly privately insured, but then he also has to deduct that in the household calculation.

So I am a civil servant and would earn 72,000 gross after the birth. With 3 children, we could theoretically earn 90k to receive the subsidy.

Are child benefits and parental allowance included in the annual income? If yes, then of course we would be above that...

Google taxable income. This is what it's about. Parental allowance, for example, is subject to the progression clause. As a civil servant, you have to have private insurance at 20% (with 3 children). You need to take this into account in your household budget.

If a property is feasible, what would be the maximum amount possible with my income without overburdening me financially?

A payment amount equal to your current rent.
There is nothing left over so far.
Another suggestion would be to rent a house of the necessary size, which is also possible without equity.
Ask tenants why they rent, one answer is always, where are we supposed to get €100,000 equity for our own house.

I will soon earn just under 72k gross income without child benefit.
I may earn a maximum gross income of 80k to receive the home ownership subsidy.

I am not clear from your posts whether child benefit and parental allowance are included in the income for the new home ownership subsidy?

If I understood correctly, then child benefit and parental allowance should not count and we would be eligible for support. If not, please clarify.

"The subsidy is exclusively aimed at families with at least one minor child in the household and a maximum taxable annual income of 60,000 euros. For each additional minor child, the income limit increases by 10,000 euros. Subsidized loans from 140,000 to a maximum of 240,000 euros are planned."