I didn't say "push" I said "encourage to explore" and I didn't mention any kind of cut off age where educators would stop evaluating kids, this would have to be a continual process not the "GATTACA" style career predetermination you fear.
Realistically, there is a segment of the population that will never be able to acquire skills that good coders have, or have acquired, regardless of how much you try to teach them. The "Hello World" crowd. I went to university with some of them.
If you stop and REALLY think about it "everyone should learn to code" makes about as much sense as "everyone should learn to repair their own car". Would everyone like to be able to repair their own car? Sure. Would this be a useful skill to have? Sure. Is everyone capable? Absolutely not.
How about this.. go and spend a few hundred hours to acquire a skill you may never use again. It has to be one with which you have had no previous exposure or proficiency. Sounds a bit stupid when it's put that way doesn't it? That's essentially what "everyone should learn to code" really sounds like.