Desktop: task chips (spawn_task) inject their prompt into the open thread instead of spawning a new session

Expected (per the desktop docs): clicking a spawned task chip starts that work in a new session with its own worktree, leaving the current session uninterrupted.

Actual: with the filing thread open, clicking the chip injects the chip's prompt as a user message into that same thread and the assistant answers in-thread. No new session or worktree is created.

Related: chips are visible only inside the thread that filed them — never on the home/sessions screen — so no click path produces the documented behavior at all.

Repro:

1. In any desktop session, have Claude call spawn_task with any title/prompt.
2. Chip appears top-right.
3. Click it while the thread is open.
4. The prompt lands in-thread as a user message.

Impact: silently defeats the feature's purpose — in our workflow it contaminated a code review that depended on a fresh session, and the reviewer announced it had inherited the build thread's context.

Environment: desktop app 1.44121.4, CLI 2.1.220, macOS 15.5. (Filed here because /feedback is disabled by our organization's policy.)