I'm fine with letting everyone try to program. Everyone should try to paint, write, make movies, play music, all those good things.

But if nobody tells someone they're stupid after they've proved repeatedly to be stupid--if we, as a culture, are too nice to leave bad reviews of bad work; if we overlook that time that Bob's excel sheet cost the company five days of downtime, and how we had to hire three extra interns to do redundant data-entry for it--then Bob might think people should be paying him a professional programmer's salary for his time. Bob might put out his shingle as a programmer. And now, the market has one more lemon.

There are some things which require real, natural (or nurtured-in, at least) talents. Singing, for example. Everything I've experienced in dealing with other programmers tells me that programming skill, in the end, is just an outgrowth of the ability to think logically, systematically, and formally--and that these abilities are part of your mental architecture, and, if they're not determined genetically, at least can only be developed when you're young. By the time someone comes into their first high-school programming class, they already will or won't be an Engineer by mindset, and there's no switch you can flick on them, no number of facts and rules you can teach, to turn that around.

To use a slightly-sour analogy, it's like the cases of children found surviving in the wild, and taken into society. They can learn words, in the same way a chimpanzee does, but they never become able to grasp syntax--their mental architecture has already set, and that component wasn't included. Logical/formal/systems thinking seems to be like this. If anyone needs to teach it, it's parents, and probably around the same age as reading. But we literally don't know what it is that's needed, specifically; what exercise you can do with a kid to "induce" logical thinking. What did we do when we were young? Play with lego? Play pretend?

I'm not completely sure I agree with you yet, but this argument does mesh very well with the trope that so many programmers first started programming when they were very young. I was playing with BASIC at age 5, and anecdotally I've heard very similar stories from other programmers at a rate that exceeds what my expectation would be of a random distribution, and definitely exceeds corresponding anecdotal evidence from other professions (i.e. number of people in profession X who were doing X at a very young age).

The point of possible disagreement is that I'm still inclined to think that any person can learn how to program at any point in time, but that it is just exceedingly uncommon for them to actually do so. Typical CS classes are certainly not going to accomplish it. Like you said, it's about thinking logically, systematically, and formally. I think very few people actually try to learn how to do that late in life. If you don't already have it, you probably don't value it enough to try to get it. It's almost tautological: how would someone who doesn't think rigorously be convinced of the value of thinking rigorously?