perf(sessions): archive a worktree's sessions through one server batch

Removing a worktree archived its sessions one SDK call at a time and
then re-rendered the whole sidebar once per streamed session.updated
echo. On a worktree with 121 sessions that meant 14.8s of main-thread
work, 121 requests, and 328 localStorage writes.

- Add POST /api/openchamber/sessions/archive: validates a batch (max
  500 ids, per-request archivedAt), archives sequentially, and reports
  partial failures instead of dropping the batch. VS Code serves no
  such route and answers 501; the shared UI then falls back to the
  per-session path.
- Plan batches from the sessions this client actually holds, live
  directory stores first, so worktree-only sessions still batch.
- Claim (id, archivedAt) pairs before the request and consume the
  matching session.updated echoes, so the server's own confirmations
  no longer fan out into 121 store publications. Runtime-scoped, TTL
  30s, released on response or fallback; non-matching updates pass.
- Make the managed-chats persistence a real trailing debounce instead
  of a 50ms throttle, so a burst of publications coalesces into one
  localStorage write.

Benchmark (121 sessions, production build, real Chrome): 14785ms ->
~1030ms, long tasks 100 -> 1, global store publications 236 -> 1,
persistence writes 328 -> 3.

packages/ui/src/sync/persist-cache.test.ts

@@ -92,6 +92,16 @@ describe("persisted directory sessions", () => {
     expect(readManagedChatSessions()).toEqual([])
   })
 
+  test("coalesces a continuing burst into one trailing session write", async () => {
+    persistSessions(directory, [session(1, 1)])
+    await new Promise((resolve) => setTimeout(resolve, 30))
+    persistSessions(directory, [session(1, 2)])
+    await waitForPersistence()
+
+    expect(storage.writes).toBe(1)
+    expect(readDirCache(directory).sessions?.[0]?.time.updated).toBe(2)
+  })
+
   test("keeps the 50 most recently updated sessions across restart reads", async () => {
     const sessions = Array.from({ length: 60 }, (_, updated) => session(59 - updated, updated))