AceParty: So good news, I managed to play via gaming mode and in portable Elite force 1 & 2.

Basically use the Widescreen and resolution files mentioned previously (on the pc wiki etc.). Be sur to use lutris to install and run the game from your GOG account.
Then, add Lutris to your steam library via desktop mode.
Launch the client via steam gaming mode (and in portable mode) once done, just click on EF1 or EF2 and enjoy! You can even use community layout. In fact, I created one for Elite force 1 and 2 with an inventory menu for the left touch pad. They actually looks really good on the small screen like most old games. Set phasers ro frag!

strubdog: Is it possible to get it working as it's own item in gaming mode without adding Lutris. Not the worst thing in the world, just curious.

Thank You for finding the workaround.

AceParty: I managed to run DS9 the fallen and bridge commander in gaming mode, so it's not that bad :D

vicofages: I searched around and found a solution that worked for me in a reddit thread that I can't link to here for some reason. Credit for this solution goes to Reddit user rebelbumscum19.

vicofages: I attempted the process outlined above and unfortunately I still couldn't get EF to run properly using Lutris.

-Using the Hor+ resolution files causes the game to display with buggy visuals and it hangs at the upside Federation logo where the LCARS main menu should be.
-Launching the game without the Hor+ resolution files loads the intro movies successfully, but it still hangs at the upside logo instead of displaying the main menu.

I searched around and found a solution that worked for me in a reddit thread that I can't link to here for some reason. Credit for this solution goes to Reddit user rebelbumscum19.

Heroic Launcher process

1. Install Heroic Launcher via desktop mode. It is found in the Discover app.
2. Log into GOG account that owns EF.
3. Install EF.
4. Select Wine Manager in Heroic Launcher.
5. Install Wine-7.2-GE-2.
6. Select EF in the Heroic Launcher library.
7. Select Settings at the bottom of the menu.
8. Change Wine version to Wine-7.2-GE-2.
9. Return to main menu for EF.
10. Select Tools at the top.
11. Make sure the game is added to Steam. If you see "Remove Steam" then it is already in Steam, otherwise select to add it to Steam.
12. Return to Gaming Mode.
13. Find EF in the non-Steam game section. Heroic Launcher will add it with the correct title and art.
14. Launch the game.

I also highly recommend AceParty's controller layout.

1. Select EF in the Steam library.
2. Select controller settings.
3. Select the layout.
4. Navigate to Community Layouts.
5. Select the layout created by AceParty (the other layout had an issue with mapping on the left stick).

So the whole thing seems to rely on a good version of Wine for the game. I didn't go back to try the improved resolution mod because I'm happy to play in 4:3 and I wanted to just get into the game, but that could also work once the compatible version of Wine is installed in Heroic Launcher.

Edit: Oh but this seems to only work in handheld mode. Connecting it to the dock leads the game to crash with an error message about OpenGL. This error message appears when the game is launched in a resolution higher than 1024x768 so it's not surprising it doesn't play nice with a larger display.