- Add _find_beads_db_in_tree() to walk up looking for .beads/*.db
- Update _get_client() to auto-detect when workspace not set
- Matches CLI behavior (no manual set_context needed after restart)
- Add 8 comprehensive tests for auto-detection
- Update existing tests to mock auto-detection in error cases
Fixes silent failures after Amp restart.
Amp-Thread-ID: https://ampcode.com/threads/T-c47f524d-c101-40d5-839a-659f52b9be48
Co-authored-by: Amp <amp@ampcode.com>
