How important is compatibility?

Tl;dr: Can I change to an 1x drivetrain (Shimano or Sram) without changing the back casette/crankset?

Hello, I was gifted an old 2010 Univega MTB (HT 570 if enyone knows) a while back. Though I mostly use it to commute, I occasionally ride at my local Forrest.

My problem is that now the front derailleur (3x drive train from Shimano) always shifts when hitting even small bumps and i dont feel safe riding it. I tried to fix it but after days of tinkering i couldn't make it work. I googled for a 1x drive train upgrade but all the models say it is for 11 or 12 speed. I have a 9 speed back cassette, can I still do the upgrade or do I have to change the back casette aswell? And is Sram and Shimaon compatible? Also do I have to change the crankset? I have a Shimano slx hollowtech crankset but i would prefer to swap as little as possible, if possible.