Edit: I'm an idiot. I made a small test with Electron and a sample webpage, and I had the same problem. Only then I realized that for some reason, unlike my browser, both Electron and WebView2 recognised my gamepad as player 2 (I'm using a "Twin USB Joystick" adapter). I plugged the the gamepad in the other slot and it worked! So, yeah, it worked!!! <3
Old message:
Ok, I'm back. I tried it on Windows and I can't get the controls to work. To be fair, my setup is far from standard, I use an original PSOne Dual Shock controller with an old USB adapter, but it DOES work with the itch version. For what I see, what Construct did was to compile your game using Microsoft Edge WebView2, so maybe it is related to this issue.
Maybe if you take your web export (the one you uploaded to Itch originally) and repackaged it using Electron instead, gamepad recognition would work better?
As a side-note: using Wine or Proton (the compatibility layer used by Steam Decks) didn't work for me on my other machine (the Linux one). But that's almost anecdotal, I don't expect Linux support.