It seems to be due to long paths. As to how and why, I am unsure.
I can confirm it's not due to the game being on a secondary hard drive compared to the home folder.

**MAKE SURE STEAM IS CLOSED DOWN DURING THE ENTIRETY OF THIS STEP**

Open this file (it should be located in the steamapps folder of your libraryfolder where Crash Bandicoot is installed):

"appmanifest_731490.acf"

Look for

"installdir" "Crash Bandicoot - N Sane Trilogy"

which should be found on line 7.

Change it to something you can still recnogize, but which is way shorter. I used "crash".
You can always return to this guide later to grab the original installdir name if you need to restore it.

Now, just save the file as is (do not change anything else in it). Then, change the folder name of Crash Bandicoot - N Sane Trilogy in the "common" folder to the exact same thing you entered for installdir.

Now boot up steam, launch Crash Bandicoot and presto, if everything went well. The game should now load levels as normal, allowing you to play the game.

For reference, here's my before and after of my path:

/run/media/xxxxxxxxxxx/SteamGames/slibrarysmoll/steamapps/common/Crash Bandicoot - N Sane Trilogy/

After:

/run/media/xxxxxxxxxxx/SteamGames/slibrarysmoll/steamapps/common/crash/

And that's it!
Why are you still here and not git gud? Go!