[Tracking issue] Improve GitHub authentication via GitHub CLI #957

There are many areas where the experience authenticating to GitHub via gh can be improved. These are the problems we'd love to solve, and the scope of this effort may be decreased or increased as we get into working on the solutions:

1. The machine I'm using with gh doesn't have a web browser available, and I need a more elegant way to authenticate that doesn't involve visiting a web browser from the same machine.
2. I'm not able to easily sign out of my GitHub account, and either reauthenticate with the same account or sign into another account.
3. gh doesn't handle Git authentication for me. For example, if I use gh repo clone and I'm authenticated with GitHub CLI, but not with Git, I'm still going to be prompted to authenticate.
4. The auth token is combined in the same config file that the rest of my settings are in, and is stored in plain text.

Related issues:
Auth without browser: #297
Sign out and back in: #887
Print URL for authentication: #612
Expired token with no way to reauth: #627
Auth failed because can't access localhost: #773
Don't store auth token in plain text: #449
Multiple account support (more elegant solution likely not in scope for this issue): #326