I played desktop Linux version, worked well for me. I couldn't use web version, there's just black screen after pressing play.
Since you used Godot 4, you may try settings `SharedArrayBuffer support on game's edit page` if you haven't already.
You mentioned in the description what you'd like to add to the game. One thing I missing from the list, I think most important - to playtest and fix the bugs and corner cases. In my opinion it's much more important than adding animations, scenarios or even tutorial.
I would often get stuck at the climbing animation and would have to restart the game. Sometimes I was stuck and couldn't do anything else. Sometimes the game would bug and let me climb the sky or any game surface, essentially disabling gravity.
Visuals look great, I would keep them the way they are now and focus on fixing bugs and improving user experience.
Also, a nice touch with custom mouse cursor. Although at first I thought that it would be used for gameplay and I tried clicking on the wall to see if anything happened :)
Well done overall, congrats!