The urge is still there to have kids. Not just to have sex but to protect the child. Ever had a girl pregnant?
The desire may exist to have and protect kids but without knowing the connection between sex and babies, you can't have willfully act on your desires. Also, the desire to have and protect kids usually manifests from the hormones of having started that process. Like you say, pregnant girls aren't overprotective before they're pregnant.
As I have said, the urge to pass on our genes manifests in many ways, and you will never convince someone not to pass on their genes because you see them as unfit.
Who says I would be deciding? Why not the parents themselves, in this possible ideal future?
Urges are associated with behavior. If the urge isn't the act itself, and you have no knowledge/rational connection to the satisfy the goals of the urges they don't influence behavior. Someone who wants to have kids but doesn't know sex will help him achieve that goal is not more likely to have sex. Did you know that monkeys who have not seen sex will not have sex? There's a disconnect where the lack of knowledge of sex from observational learning inhibits their behavioral desires. In these cases, they actually tend to just avoid the females entirely. Of course if you can eliminate the capacity for knowledge by converting it to pure instinct (not goal oriented) you can circumvent learning obstacles. Rats for instance will mate without previous knowledge of sex.
The caveman with no knowledge of the causality behind sex will not have an urge for sex beyond that of sexual stimulation, but just a frustrating desire for kids that he doesn't know how to satisfy.
Perhaps someone will be unwilling to procreate because of a disease they have that would be passed on but even that is rare.
Unfortunately evolution tends to work against higher order logic. Those willing to sacrifice their urges are actually what we need in the gene pool. The largest hope I can think of is enlightening the society through education.