Apple TV turns off TV when it times out

This is not so much a question as a "I found this so you don't have to".

We recently replaced our ancient Apple TV box with a new Apple TV HD, and connected it to our ancient LG Dumb TV which had served us so admirably for many years, but has no smart platform (necessitating the Apple TV box). It is still serving us admirably, in fact, but that's another arc in the story. With 3 HDMI devices and only 2 HDMI inputs on the The, we had bought an HDMI switch (with remote!) for easy switching between inputs and routed the cable receiver, Apple TV and DVD player all through that. This setup worked swimmingly for years. But with new security vulnerabilities becoming a daily news item and the ancient Apple TV box being unable to run the latest tvOS, it seemed prudent to replace it.

Shortly after we set up the new Apple TV HD (with the latest tvOS and the latest version of all of the apps we commonly use, huzzah!), we started having what appeared to be an intermittent problem with the TV suddenly turning itself off without warning or explanation. We could turn it back on again, and it would operate fine -- until it turned off again. Thinking our trusty LG was finally breathing its last, we replaced it with a new somewhat smart Samsung that was smart enough for us to watch, but not the other way around.

This did not fix the problem.

In other words, it was not the TV. Our brand new Samsung was doing the same thing as our old reliable LG. Which is now relegated to the second bedroom where it will pine for its glory days, getting only occasional use because cats don't have thumbs.

Next oldest item in the chain was the HDMI switch, which was a cheap one I'd found online, but had done the job for several years. This was replaced by a shiny new 5-port hub that was a significant (but very reasonable) upgrade.

This also did not solve the problem. But this wasn't immediately obvious because there were no hockey games for several days, so we were watching the Apple TV almost exclusively, and as long as the Apple TV was on (or definitively off), there wasn't a problem.

Thinking that the surge suppressor everything was plugged into was potentially the culprit, we swapped it out for a different one. Which also appeared to work for a while, because the actual root cause was able to disguise itself fin such a fiendishly clever way.

It took about 2 weeks, and a lot of trial and error, and false hope, but eventually one of us (not me) noticed that when the TV spontaneously powered down, the new Apple TV was also turning off.

AHA! Come, Watson! The game's afoot!

Normally it would have occurred to me to examine the thing that had changed as a root cause. But because of a weird convergence of circumstances, and the intermittent nature of the problem, it looks a month or more to even realize that there was a problem. By which time, the new Apple TV was no longer "new", and since there wasn't any obvious problem with it, and the other components were almost 10 years old, they got blamed.

There is an older thread from 2017 reporting the problem, but didn't really offer a solution, just a link to the support article If you can't control your TV or receiver with your Apple TV Remote - Apple Support (CA)

That's not the use case in the question (though it did give me a general idea where to look for the offending setting. Which I will detail in a response so that those who just want to skip to the end and not read my pithy commentary can do so and get the answer without the question.