extension login with non-default browser and remote vscode mode has no effect

## Problem

I can't figure out how to get the extension logged in so the explorer works

## Steps to reproduce the issue

1. Clean slate
2. install firefox, make default browser (perhaps unrelated, IDK).
3. install chrome
4. create a default chrome profile
5. create a second chrome profile for 'work', log into AWS identity centre in that profile
6. create a Hyper-V VM, install ubuntu
7. inside that VM install aws cli v2.current and run `aws configure sso`, verify it works
8. install vscode, add the remote extension, connect to the VM as a remote folder
9. add vscode AWS extension, click on the

10. Copy the code and put it aside
11. Select 'copy' when given the URL, rather than open, because the default browser doesn't have AWS credentials.
12. Paste the URL into the chrome work profile
13. paste the stashed code into the prompt
14. approve the request
15. close the browser window
16. observe that the VS code extension still says it is expired or invalid

## System details (run the `AWS: About Toolkit` command)
OS: Linux x64 5.15.0-1033-azure
Visual Studio Code extension host: 1.76.1
AWS Toolkit: 1.64.0
node: 16.14.2

Reproduce the login flow in the issue using VS Code Remote and a non-default browser profile, starting with the AWS Toolkit login action and the AWS: About Toolkit system details. Confirm whether approval leaves the extension expired or invalid and whether the explorer remains unavailable; done means the remote extension recognizes the completed login and the explorer works.

Written by the indexing model from the issue text.