Hi,
I am experiencing the exact same problem. I also use Gnome Shell, and the browser doesn't matter - I've tried Chrome, Chromium, Firefox, and Vivaldi.
Sometimes however it does work, but this is usually when I have just logged in. Then it will sometimes work a few times, and later it no longer works. It's as if something is blocked at some point, preventing the assistant from coming up. This is supported by the fact that I do see the assistant briefly flashing up and then vanishing again.
Please let me know if I can help diagnose the problem.
Thanks.