I think the point here is that the people who don't want kids are still a useful part of our gene pool, and we should look critically at what factors in to the decision not to have any kids - I don't think it's genetic at all.

It's impossible for me not to think of Idiocracy while writing this - which was extremely exaggerated, but still...

Are you thinking of memetics?

Astute observation, but given the fact that we also seem to be seeing an increase in fatherless/split families (indicating poor choice in parenting partners and decisions) it's not likely to simply see-saw back in necessarily healthy ways.

That's not how natural selection works. Most of us want sex because of that gene. But we do know about contraception, which natural selection knows nothing about.

Contraception is part of the environment in which natural selection occurs, in that sense it "knows" about it. Concretely, if there is a set of genes that makes it more likely to have children in spite of contraception, that can be selected for.

No. You do not pass e.g. ability to read via genetic code. So you do not pass knowledge about the contraception (or lack of it) either.

If reading is advantageous to select for, you'll find that natural selection will prefer whichever genes facilitate reading ability

There are a lot of subtle hereditary hints. This isn't to say a person can't consciously counteract those hints, but that in the long run those hints will have an impact on behavior which feeds back into selection

It's possible to inherit traits which will hint one to choose not to use contraception

This is the whole nature vs nurture debate. Ultimately it's both

That assumes there is such a thing as a "I want kids" gene, which I personally have never heard of.

Not necessarily one gene, per se. More like a set of characteristics and pre-dispositions that make kids seem fun and desirable will be selected for, while the opposite will be selected out.