Sessions archive/drop unexpectedly and scheduled tasks don't sync to iOS remote control

**Summary**

Two related symptoms with the remote-control feature:
1. Scheduled/local tasks that run correctly on the desktop app do not appear in the iOS remote-control app.
2. Existing sessions get archived or dropped unexpectedly when using remote control, which is disruptive to ongoing work.

**Environment**
- Claude Code version: 2.110.0
- macOS 14.8.9 (build 23J631)
- Claude Code desktop app (Code tab), remote control enabled (`remoteControlAtStartup: true`)

**Reproduction / evidence**
- Scheduled tasks: 8 tasks are registered locally (via the scheduled-tasks system) and run correctly on schedule with recent `lastRunAt` timestamps (e.g. daily commit-queue reminder, weekly improvements review). None of these show up in the iOS remote-control view.
- Session archiving: sessions have been observed archiving/dropping unexpectedly during normal use.
- Ruled out local causes: a third-party Claude Code hook tool (running background daemons that periodically restart sessions to manage context) was fully disabled for 2 days (all hooks removed from `~/.claude/settings.json`, all its background processes killed) and the archiving/dropping issue still occurred. This rules out local hook/automation interference as the cause.
- This has been observed intermittently over roughly two weeks (first noticed ~2026-09-05, still occurring as of 2026-09-16).

**Expected behavior**
- Scheduled tasks registered on desktop should be visible in the iOS remote-control app.
- Sessions should not be archived or dropped without explicit user action.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Start with the remote-control feature and the scheduled-tasks system, using remoteControlAtStartup in ~/.claude/settings.json to reproduce both symptoms. Compare the desktop task state with the iOS remote-control view and observe session behavior with third-party hooks disabled; done means scheduled tasks appear remotely and sessions remain intact without explicit archiving.

Written by the indexing model from the issue text.