This sounds like to me as though it's essentially a giant recapitulation of the trolley problem - you have one side claiming that the opposition doesn't understand 5 > 1 and isn't trying to maximize utility and should be pushing the fat man onto the tracks, and you have the other side not wanting to violate obvious moral norms such as "Don't push people onto train tracks where they will die" for the sake of hypotheticals that are not merely unlikely but unrealistic. (How is that man so fat that he'll block a trolley going fast enough to crush five other people, anyway?)
Then the first group argues that sometimes you need to be able to engage rational overrides when the situation is different from what you're used to (or adapted to) and in this case we're stipulating that the man is fat enough to stop the trolley if you push him onto the tracks, and the second group argues that you want bright-line ethical rules and guards against corrupted hardware and con artists trying to convince you to do evil deeds "for the greater good".
In this case a bit less freedom would guarantee a lot less distress.
If this statement is to be taken as a hypothetical stipulation similar to that of the trolley problem, I agree with the hypothetical breeding license.
As a real-life policy suggestion, though, it sounds like a terrible idea due to violating a lot of people's moral norms (which will cause distress), having implementation difficulties (who will make/mark the tests for getting a license), being prone to frighteningly nasty abuse, and underspecification. Please do not take my support of the hypothetical stipulation as being in any way supportive of the actual policy suggestion.
Sir, Could I See Your Breeding License?
Most important is that children don't have to grow up under horrible circumstances inflicted on them by the inability of their parents. You always have to weigh the freedom of some against any negative infliction it could have on others. In this case a bit less freedom would guarantee a lot less distress.
It is reasonable. I don't see how we can ask for species-appropriate animal husbandry regarding animals like chimps but not children. You have to have a drivers license for good reasons too. So why is everyone allowed to rule over helpless human beings for years without having to prove their ability to do so in a way that guarantees the well-being of their protégé?
Such discussions always remind me about something important. Children should not be assigned with any religion. There should be a certain age where they can decide what religion they want to follow, if any. This doesn't mean that religious people shouldn't be able to have children but that they shouldn't be able to force their children into a certain framework either. Parents should be forced to allow their children to take part in a educational framework based on contemporary ethics and knowledge. I don't even have a problem with lessons in religion in school as it is part of human nature. But it shall not be focused on any truth value or a certain religion but an overview and comparison with non-religious ethics and truth-seeking.