Hellooo! First just to clarify: 3rd party auth is supported for Free and Individual plans. Additionally, how you prefer to sign in to your account does not affect how keys/tokens connect to your databases – we can trust that your account login through Github/Netlify is not affecting the authorization of your users.
You are right that you can get unauthorized errors if you do not specify the correct Region Group endpoint (See the Connections page for your preferred driver). If you are certain that the domains are set appropriately, then the roles are the next thing to look at.
You can run a simple query that fetches no data to determine if your connection is properly authorized. The following should need no permissions, so if it succeeds then you can know the connection is configured properly.
client.query(q.Abs(1))
If this succeeds, but your other queries do not, then there is likely permissions missing from your Role.