Not getting logged in when setting GITHUB_TOKEN env

### Is there an existing issue for this?

- [x] I have searched the existing issues

### OS/Web Information

- Web Browser: Chrome 133
- Local OS: win11
- Remote OS: ubuntu 24.04
- Remote Architecture: amd64
- `code-server --version`: 1.96.4

### Steps to Reproduce

1. Create workspace
2. mounted home directory
3. run the following in startup_script
4. `export GITHUB_TOKEN=<>`
5. `/app/code-server/bin/code-server --auth none --port 13337`

### Expected

Be logged into Github Actions, Pull Requests, etc

### Actual

Getting asked to sign in with Github to use *

[main-logs.txt](https://github.com/user-attachments/files/18711096/main-logs.txt)

### Logs

```shell
look above it was too big
```

### Screenshot/Video


### Does this bug reproduce in native VS Code?

This cannot be tested in native VS Code

### Does this bug reproduce in GitHub Codespaces?

This cannot be tested in GitHub Codespaces

### Are you accessing code-server over a secure context?

- [x] I am using a secure context.

### Notes

_No response_