My set-up is a Mac, Apple-Silicon, Visual Studio Code 1.93.1 and "Raspberry Pi Pico" v0.16.2.
python3 is installed via homebrew and can be executed in the terminal. I even provided the full path to python3 within the settings of the extension. I am not sure where to find any log information by "Raspberry Pi Pico" so I get stuck here.
Interestingly, if I uninstall the extension, remove any cached downloads, and re-install, the first time using "Project from Example" works. Afterwards it is not working anymore.