Desktop Browser pane has no way to use existing browser login state — no profile/cookie import, and the Chrome extension doesn't fill the gap

## What's missing

The Desktop app's built-in Browser pane runs on a clean browser profile with none of my logins. There is currently no way to bring an existing browser session into it — no profile import, no cookie import, no opt-in "use my existing profile" mode. The docs state this is intentional and point to the Claude in Chrome extension instead.

That recommendation doesn't close the gap:

1. **The extension drives my everyday browser** — my tabs, my window, my focus. I don't always want the agent operating in the browser I'm personally using.
2. **The extension effectively serializes to one session.** Multiple Claude Code sessions can't safely share it concurrently, so additional sessions have to wait their turn. The Browser pane *is* per-session and parallel-safe — but it's the surface that can't authenticate. The capability I need for parallel work sits on the wrong side of the split.

## Prior art

The ChatGPT/Codex desktop app already ships exactly this. Its built-in browser shows a one-click **"Import data from Chrome — Bring over your passwords and cookies to the built-in browser."** The pattern is established, users coming from that app expect it, and its absence here is felt immediately.

## What I'd like

An opt-in way to give the Browser pane real login state. Any of these would solve it:

- Import cookies/profile data from an installed Chromium browser on first use
- Point the pane at an existing browser profile directory
- A persistent, user-managed signed-in profile for the pane that survives restarts and is reusable across sessions

Opt-in, clearly scoped, revocable, and off by default is completely fine. The point is that it should be possible at all.

## Why it matters

Anything behind a login — dashboards, admin consoles, internal tools, paywalled docs — is out of reach for the Browser pane today, and that's a large share of real work. The current options are signing in manually inside the pane for every site, or falling back to the extension and giving up parallel sessions. Closing this would make the built-in browser usable as a primary surface rather than a preview-only one.

## Related

Adjacent but distinct: #80293 and #83163 (pane loses cookies across restarts), #82668 (browser automation cannot authenticate). This request is about *inheriting existing* login state rather than persisting state created inside the pane.

Begin with the Desktop app's Browser pane and compare its authentication behavior with related issues #80293, #83163, and #82668. Define one opt-in, revocable import or profile flow that supports persistence and parallel sessions, then verify that login state survives restart without changing the default clean profile.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.