replace gmail resync with Gmail REST API, remove mbsync dependency

First sync still uses IMAP with app password. Subsequent syncs use
Gmail API history.list + messages.get with OAuth for faster, more
reliable incremental sync. Dispatch gmail-sync handler for gmail
accounts instead of generic email-sync. Show sync button for synced
accounts.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

src/servers/api/email/accounts.ts

@@ -47,7 +47,7 @@ accountsRouter.get('/', async (ctx) => {
       const jobs = await listAllJobs();
       activeJobAccountIds = new Set(
         jobs
-          .filter((j) => j.type === 'email-sync' && (j.status === 'queued' || j.status === 'running'))
+          .filter((j) => (j.type === 'email-sync' || j.type === 'gmail-sync') && (j.status === 'queued' || j.status === 'running'))
           .map((j) => (j.meta as Record<string, unknown> | undefined)?.emailAccountId as number)
           .filter(Boolean),
       );
@@ -148,9 +148,11 @@ accountsRouter.post('/:id/sync', async (ctx) => {
   // Set status immediately so the UI reflects the queued state
   await updateEmailAccountStatus(id, 'queued');
 
+  const jobType = account.provider === 'gmail' ? 'gmail-sync' : 'email-sync';
+
   const job = await enqueueJob({
     lane: 'email',
-    type: 'email-sync',
+    type: jobType,
     userId: user.email,
     meta: {
       emailAccountId: id,