Unless the game is solely bottlenecked by processing stuff

yyyyep. Pretty much any mainline GPU made since 2010 should be able to handle TF2 just fine, and if you've got a 1060 or better I seriously doubt your GPU is the problem. The problem is that CPU usage is horribly optimized and it can only really utilize 2 threads (not cores, threads). After that it's your clock speed that makes a difference.

I played competitively on a 680 and an overclocked i5-4690k@4.2Ghz until I finally upgraded last year, and would only dip below 100FPS playing pubs on Halloween. In 6s I never went below 200.

I mean, back in the day, I already used to get better performance when I would boot in Ubuntu 16 instead of Windows. Not sure that's the case anymore with modern stock Ubuntu but I imagine Mint would still do better thanks to having less bloat. But I have trouble imagining that better shaders are going to help many people at all.

edit: the vulkan update probably won't help. the switch to 64 bit has the potential to be HUGE.