I was going to comment on gamepad too -- it looks from the comments like you discovered the same thing I did that mapping is kinda screwed up with web games dependent on JavaScript's API especially with a switch controller. When I made Quad, I solved that by making a separate gamepad mapper to handle re-mapping all the controls since I really needed every button and both joypad axes to work like I wanted for that game, you can play it to see what I mean at
https://www.newgrounds.com/portal/view/848392
I also posted the source code for the Godot autoload to make it work, which made things as easy as I could for other devs to implement it in their own projects too.
https://3p0ch.newgrounds.com/news/post/1261146
Although it might need some tweaking to convert from Godot 3 to Godot 4.
Also IDK why saving isn't working for you but feel free to msg me if you want the Godot 4 form of the autoload I usually use for handling save data, and/or are having issues with cloudsaving with the Newgrounds API implementation.