I would prefer to understand why a paused or backgrounded game still manages to consume a ton of CPU or GPU

Like, you're still just churning away at the main game loop while literally nothing else is happening except for you waiting for me to unpause it?

Because THAT would be an actual achievement. Hell, I can suspend any process from the Unixy side of things by sending a SIGSTOP signal, for a far more perfect "pause".

If I was a game dev, I would not settle for "looks like paused, but still burning down the environment and heating up my home as a side effect"

Because the engine is still running, even in a paused state, the game still has to show something and process input. Sometimes there is a menu too, sometimes the game is not completely frozen: flames may flicker for instance.

In the article, there is a case where the game takes a screenshot and disable costly rendering, presumably to deal with this problem. But the thing is that most games are designed to to be actively played, not paused for extended periods of time and having an active pause for a couple of minutes isn't going to destroy the environment.

For backgrounding, is is common for games to run at a much slower framerate to save resources.

But if you were a game dev, you would understand why it‘s not as easy as it seems to outsiders. :)