Flatpaks can access joysticks but you have to allow access to devices. You can see that in Fight or Perish's flatpak manifest:
https://gitlab.com/dulsi/fop/-/blob/main/flatpak/com.identicalsoftware.finishorperish.yml
For linux games on itch, I tend to build on Debian and go with a zip and LD_LIBRARY_PATH. I also include a .itch.toml file. That ensure it can run directly from the itch app. I don't know how many people use the app but just in case.