[FEATURE] Let Claude Design write to GitHub and Azure DevOps repos

Problem
Claude Design can read from GitHub.
It cannot write to GitHub or Azure DevOps.
The local handoff to Claude Code is broken as well (see #69246).
So the only working path is: download a zip, unzip it, hand it to Claude Code by hand.
That breaks every automated workflow.

My use case
Claude Design builds complete frontends, including JS.
Claude Code then implements the backend and wires everything up.
Both run in the Claude desktop app.
Design and code need to live in the same repo, without manual file transfer.

Requested

1. Claude Design can commit and push to a branch in a connected GitHub repo.
2. Claude Design can open a pull request.
3. The same for Azure DevOps repos (write and read).
4. The local handoff to Claude Code works in the desktop app without a zip fallback.

Why it matters
With write access, Git becomes the single source of truth.
A push from Claude Design can trigger Claude Code via CI (GitHub Actions / Azure Pipelines).
Design and code stay in sync instead of drifting apart.