refactor: drop the vestigial bot_bottle_root/host_db_path re-exports (#352)

paths is the single home now, so stop re-exporting the path helpers from
supervise: remove host_db_path from supervise's imports + __all__ (it was
re-export-only) and drop bot_bottle_root from __all__ (kept as an import,
still used by audit_dir). supervise_types was already clean. Repoint the
last readers (test_supervise imports host_db_path from paths;
test_supervise_edge calls paths.bot_bottle_root) and refresh the doc
mentions. No supervise.bot_bottle_root / supervise.host_db_path references
remain.

Behavior-preserving: full unit suite unchanged (only the pre-existing
/bin/sleep sidecar-init errors).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WBMWTEtQdJ4W5UrWuLHCck

tests/unit/test_supervise_cli_crash_logging.py

@@ -40,9 +40,8 @@ class TestDieCarriesMessage(unittest.TestCase):
 
 
 class _FakeHomeMixin:
-    """Point supervise.bot_bottle_root (what _write_crash_log resolves
-    through) at a temp dir so the crash log doesn't touch the real
-    ~/.bot-bottle."""
+    """Point bot_bottle_root (what _write_crash_log resolves through) at a
+    temp dir so the crash log doesn't touch the real ~/.bot-bottle."""
 
     def _setup_fake_home(self):
         self._tmp = tempfile.TemporaryDirectory(prefix="supervise-crash-test.")