It really isn't part of the idle game genre, but it's perfect to turn your brain off to and can be played completely with a mouse: The Sims. It's not great, obviously, has a shitton of issues, and I get if it doesn't meet your criteria. But it does not require you to get extremely involved, has a lot of customization options (the older versions especially), and you can pretty much let it run by itself. To me, at least, it's perfect for when you listen to podcasts or audiobooks and need to keep the hands & eyes busy. Maybe that'll do for your purposes.