A "Datasource Exception Error" with google datasources typically means the Tableau Server doesn't have access to the data. You will likely need to use a Google App Password to bypass the two-factor authentication on many Google accounts. If you want to double-check, press F12 to open a developer's console in your browser, go to the Network tab and refresh the page - you'll likely see some 401 errors saying bad password. It works fine with Tableau Desktop on your client computer since you are already logged into your Google account.
Hope this helps!
But I'm not using any 2FA on this Google account that I'm trying out.
I got as far as seeing a "Transferring Data to Server" when publishing from Tableau Desktop before seeing the error.
Anything else I can try?
I would be curious if there is additional information contained in your Tableau Desktop logs that may shed light on the situation. I would suggest recreating the issue after setting up clear desktop logs (Sending Tableau Desktop Log Files | Tableau Software) and see if there is anything in the logs pointing you in the right direction. If not, might be time to open a case with Tableau Technical Support for further investigation. They'll likely want copies of those Desktop Logs anyway.
Thanks and hope this helps!