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Not my experience. I've had multiple old games and an old printer that just straight up didn't work under Windows. On Linux however (using wine for the windows exe's) it usually does run. Sometimes it does require some googling, but there's usually someone who tried it before.
Games are actually the hardcore compatibility test. They are much less compatible than the average piece of software. That's due to them using much more of the hardware/low-level-APIs of the OS, but also due to DRM and Anti-Cheat-Software (where applicable).
And printers are also (for some reason) super difficult. Probably because they are cheap, planned-obsolescence pieses of crap hardware, which are chock-full of DRM.
Shouldn't window's compatibility mode solve most of those?
Should, yes, but I find it often doesn't.