I have tried to recreate the Apps in dom0 which doesn’t work because the App isn’t in the template. If I move the App directly to the template, then I don’t think the models will store.
Although it’s not Qubes specific, I think the documentation is lacking some steps about how to check that a shortcut is working inside the AppVM. Using a XFCE template, you can try to open xfce4-appfinder and see if the ChatGPT shortcut is present. To see which .desktop file will be sent to dom0, you can run:
/etc/qubes-rpc/qubes.GetAppMenus
The content of the file seems good to me, but have you validated it?