fix(browse): preserve live daemon during busy loads

[time-attack]

During a heavy page load, browse could mistake a busy browser daemon for a dead one and restart it. That discarded open tabs, cookies, and logged-in sessions.

Browse now preserves an unresponsive daemon while its process is still alive and asks the user to retry. Automatic restart happens only after the daemon process is actually gone.

Technical details

• Treat an OS-live daemon as busy when its HTTP health endpoint temporarily times out.
• Apply the safeguard during initial daemon discovery and command connection failures.
• Preserve existing recovery behavior for processes that are actually dead.
• Add regression coverage for live, dead, and missing daemon states.

Evidence

The same live CLI probe ran in five isolated baseline/candidate repo pairs:

• gstack-auto: baseline failed; candidate passed
• zotero-arxiv-daily: baseline failed; candidate passed
• lacp: baseline failed; candidate passed
• stagehand: baseline failed; candidate passed
• plane: baseline failed; candidate passed

Every baseline killed the daemon and replaced its session state. Every candidate preserved the original PID and session token.

Secondary checks

• 12 targeted tests passed
• bun run build passed
• Version and release files were not changed

Fixes #2219

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[time-attack]

@time-attack Please rewrite daemon recovery so busy sessions remain protected while a live but permanently wedged daemon can still be replaced safely.

[capy-ai]

Thank you for this fix! The bug it addresses (#2219) has since been fixed on main and the issue is closed, so I'm closing this PR as done. Re-verified against current main (v1.91.12.0). If you think part of your change is still missing, reply and we'll reopen. Really appreciate the contribution.