Glad to see I'm not the only Pico 4 user here :P
I just ported my entry to WebXR using Snopek's export templates for Godot 4.7, and it works pretty well on my Pico 4. The basic gameplay works, the framerate is not very high though :P
If you'd like, you can test it here: https://bauxite.itch.io/build-a-lighthouse
Just open the link in the Pico Browser in the headset and it should work. (You can also open it on a Chromium based browser on your PC for PCVR, for better performance)
When I have the time I'll test other's entries on my Pico 4 to see if I have the same issues as you.