What's the consequence for being deemed unsuitable? Who determines that you are unworthy of reproduction? What other fundamental rights should unsuitable people be deprived of?
When you consider that the type of family (highly educated) that can actually afford to have children is deciding against having children then your proposal to consider every child as a hobby is just plain stupid. Who's going to have children then? Not the upper class? not the middle class? Not the poor? Nobody?
Plenty of people are still going to have children, anyway; if we at some point find ourselves short of children, we can bring the incentives back (or just have more immigration). But I don't think it's likely to happen over the long term... the days of needing to make the maximum number of factory workers and cannot fodder for the motherland are over, at least for a time. Having children is totally a hobby.
The same exact reasoning applies to having children. I don't see any good explicit reason to produce more children across the board, in fact I see the reverse (be it global warming and the per capita emissions in the first world, the recent electoral successes based on the plight of poor workers suffering due to globalism, automation, whatever); plus there's also plenty of willing immigrants that would solve the governments/societies problem, if there even was one. So, the government has no good reason whatsoever to subsidize child-making. If someone wants to do it as a hobby, all power to them!
Treat every child in poverty as a societal responsibility. Everything will stem from there.
a.k.a. start with societal empathy.
Where will you find enough qualified teachers and nursing staff as the parents are obviously unsuitable? Cooks? Farmers? State would literally have to order at least some of those people to take such jobs.
Maximum of factor is perhaps 10 per person, likely much less than that to provide proper quality. Unless you will invest heavily in teaching and nursing staff.
It is an interesting model of a society halfway ran like military. Ultimately not too bad and orderly, but the problems of allocation are hard to solve and corruption is still likely.
The old adage is that the more you want to control, the more you have to control. (Providing strong incentives might be enough.)
Mind you EU socialdemocratic model is only halfway there, and thus limited in results. (Orphanages and nursing are too few, poor people are not relieved off parenthood. Free housing and food is very limited using merit as allocation through social services.)
A perfect example is when one of the parents succumbs to addiction that wasn't present before the child.
Also, this belittles the cases where one parent was making a great choice, by leaving the other.