Custom connector handshake succeeds (200 OK) but tools never appear in tool_search

### What happened?

A custom Connector (a self-hosted remote MCP server, streamable-http,
no authentication) shows as "Connected" in Settings > Connectors, with
all 17 tools set to "Always allow." Despite this, no tools from this
Connector are ever recognized in chat — tool_search returns zero
results for this Connector's tools, while tools from other Connectors
(e.g. Notion) return normally.

The server itself has been confirmed healthy and reachable:
- Direct POST/GET requests to /mcp consistently return 200 OK with
correct responses
- A POST simulating a tools/list request returns a valid JSON-RPC
response containing all 17 tool definitions
- Inspecting the tunnel's request log (via ngrok Inspector on
127.0.0.1:4040) confirmed that Claude.ai's backend does send
requests to the server and receives 200 OK responses — multiple
new requests were logged immediately after reconnecting the
Connector from the Settings UI

In other words, the MCP handshake itself reaches the server and
completes successfully, but the resulting tool list is never
reflected in the chat session's tool index. This is not a duplicate-
Connector issue (confirmed only one entry exists with the correct
URL) and not an authentication issue (the server requires no auth
at all).

### What did you expect to happen?

Since the Connector shows as "Connected" and the server responds
correctly, I expected the 17 tools it exposes to become available
via tool_search (or the tool list) in chat sessions.

### Steps to reproduce

1. Register a custom MCP server (streamable-http, no auth) as a
Connector in Claude.ai Settings > Connectors
2. Confirm it shows "Connected" with tools set to "Always allow"
3. Start a new chat and run tool_search for keywords related to
this Connector's tools
4. Result: tools from other Connectors (e.g. Notion) return normally,
but this Connector's tools never appear — zero results
5. Reconnect the Connector (disconnect → reconnect) while watching
the tunnel's request inspector — confirm that POST /mcp requests
actually arrive and return 200 OK
6. Re-run tool_search in the same chat — the Connector's tools are
still not recognized

### Area

MCP Connector (adding/managing servers)

### MCP Server (if applicable)

Custom self-hosted server (streamable-http transport, exposed via a tunnel service, no authentication required)

### Error messages or logs

```shell
No explicit error is shown in the UI — the Connector remains marked
as "Connected."

Server-side log evidence (showing normal operation):
POST /mcp → 200 OK (7.69ms)
POST /mcp → 200 OK (4.24ms)
POST /mcp → 200 OK (2.52ms)
GET /mcp → 200 OK
(all captured via the tunnel inspector immediately after reconnecting
the Connector)

GET /mcp response body:
{"name": "mocka-memory-caliber", "version": "1.3.0"}

The only observable "error" is the silent absence of this Connector's
tools from every tool_search result.
```

### Additional context

- The server requires no authentication, so this is not an OAuth/
token issue.
- The ngrok free-tier warning interstitial was tested and ruled out
as a cause (response is identical with or without the bypass
header — always 200 OK with the correct body).
- Only one Connector entry exists for this server — confirmed no
duplicate registration.
- The tunnel endpoint was rotated once during troubleshooting; the
symptom persisted identically before and after, suggesting the
URL itself is not the cause.
- Similar symptoms (Connector shows "Connected" but its tools never
load into chat) appear to have been reported in other issues in
this repository.

Note: I filed this same report as #534, which was closed without
any comment, fix, or linked PR. The issue is still reproducing.
Re-filing since I'm not sure why the original was closed.