Add "Integrated browser" support for code-server

## What is your suggestion?

I would love to see the "integrated browser" from vscode inside code-server (not to be confused with the "simple browser" that is already integrated).

## Why do you want this feature?

The simple browser has limitations regarding localhost servers. E.g. reaching a local vite.dev webserver is not possible.

## Are there any workarounds to get this functionality today?

You could open a port and expose it via the ports suburl. But that prevents to use automation features that the new "integrated browser" has.

## Are you interested in submitting a PR for this?

Yes. I would like the team to estimate whether an integration is technically possible and to know which main knobs need to be turned.

See my issue in the official vscode repo for current technical limitations (vscode.dev)
https://github.com/microsoft/vscode/issues/331784#issuecomment-5359696592

Start by reading the linked VS Code issue and reviewing how code-server's existing simple browser and ports suburl work. Compare their localhost and automation limitations with the requested integrated browser, then document the main integration knobs and the technical feasibility; done means the scope and blockers are clear enough for a PR plan.

Written by the indexing model from the issue text.