Github MCP server: Re-Auth does not seem to work

Testing #279705

- enable `"github.copilot.chat.githubMcpServer.enabled": true`
- set 
```
    "github.copilot.chat.githubMcpServer.toolsets": [
        "default",
        "orgs"
    ]
```
- ask 'Can you find me all orgs I am part of?'

<img width="455" height="648" alt="Image" src="https://github.com/user-attachments/assets/2ccfd868-ff96-4657-b3b2-b40d2e4ee08d" />