That’s ok, because by having children you provide a service to the state (i.e. ensuring its existence in the long run).

But this argument can be trivially extended to all other "social engineering" taxes. Vice taxes make you drink or smoke less? You're reducing state's expenses on healthcare, law enforcement and social services. Tax benefits for installing photovoltaics or buying electric cars? You're helping to improve the quality of life, again saving state on healthcare, and in the future, on various forms of crisis management once the climate issues start taking their toll in full.

(FWIW, I'm generally in favor of this kind of "social engineering" when done by democracies.)

Yes, Why should a person the choose not to have children be punished by the state with higher taxes for their choice?

If having children benefits society those who choose to do that should be rewarded.