bug: integration list shows 'syncing' forever after initial sync completes (status never flips to ready)

[khaliqgant]

Symptom

After Slack connected + initial sync completed, relayfile integration list still shows:

provider   status    lag   last_event_at
github     ready     0s    2026-05-28T22:30:46Z
slack      syncing   0s    -

slack is stuck at syncing (and last_event_at empty) even though the sync is done. Should be ready.

Likely root cause

The sync-status reporting never transitions the provider to ready after the initial sync finishes (or the readiness signal isn't persisted/propagated). Possibly the same conflation seen on the cloud side (cloud#1406): a provider can be OAuth-connected + synced but the status route keeps reporting a non-ready/pending state. github flips to ready, slack doesn't — worth diffing why (provider-specific sync-completion signal?).

Ask (fix comprehensively — relayfile and/or cloud)

• integration list status should reflect true readiness: ready once connected + initial sync complete; syncing only while actually syncing.
• Reconcile with cloud#1406 (deploy connect-check reads sync-ready instead of oauth-connected) + relayfile#217 (connect blocks on sync) — these are one cluster of integration connection/sync status-reporting bugs.

Found during overnight pr-reviewer E2E (slack on workspace 50587328…).