Scheduled routines (RemoteTrigger) report 'no connectors' despite connectors showing as Connected in claude.ai

## Description
When setting up a scheduled cloud agent (a "routine", via the `/schedule` flow), the connector-availability check consistently reports no MCP connectors available, even though the account's connectors page (`claude.ai/customize/connectors`) shows multiple connectors in a "Connected" state (Gmail, GitHub, Google Drive — all with a checkmark under "상태").

## Steps to reproduce
1. Connect one or more connectors (e.g. Gmail) at `claude.ai/customize/connectors`. Confirm they show as Connected in that UI.
2. Start `/schedule` (or otherwise query available MCP connectors for a routine) in a Claude Code session.
3. Observe the "Available MCP Connectors" section.

## Expected behavior
The connected services (Gmail, GitHub, Google Drive) should appear as available connectors that can be attached to a routine via `mcp_connections`.

## Actual behavior
The check repeatedly returns "No available MCP connectors found." This was re-checked multiple times over roughly 1 hour after the connectors were confirmed connected in the UI, with no change.

## Impact
This blocks attaching any claude.ai connector (e.g. sending an email via Gmail after a routine finishes) to a scheduled routine, since routines can only use claude.ai connectors and the connector list never reflects what's actually connected.

## Environment
- Feature: Claude Code scheduled routines (`RemoteTrigger` / `/schedule`)
- Connectors confirmed connected: Gmail, GitHub, Google Drive

The relevant entry points named here are `/schedule`, `RemoteTrigger`, and `mcp_connections`; begin by tracing the connector-availability check after confirming Gmail, GitHub, or Google Drive is Connected at `claude.ai/customize/connectors`. Done means a connected service is reported as available and can be attached to a routine through `mcp_connections`, with the reported reproduction covered by a test if the existing test area is identified.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.