[BUG] claude.ai web: lemlist community connector fully healthy in Settings (58 tools listed) but zero tools available in chat, persists after full OAuth re-authentication

### What happened?

The lemlist community connector (remote MCP, https://app.lemlist.com/mcp) reaches a fully healthy state on claude.ai web: status "Connected", Settings page lists all 58 tools (4 interactive + 54 read-only) with correct titles and per-tool permission controls set to "Needs approval". But in chat, including brand-new conversations started after setup, the model reports that no lemlist tools are available. Other connectors on the same account (Make, Netlify, Google Drive) load normally in the same conversations.

Already attempted, with no effect:
- Opening multiple brand-new conversations after setup
- Disconnecting the connector and completing a fresh OAuth flow (lemlist consent page, full API access granted, callback to https://claude.ai/api/mcp/auth_callback succeeded)
- Verifying the server is alive: GET on the MCP URL returns the service manifest (transports HTTP+SSE and Streamable HTTP, OAuth metadata with Dynamic Client Registration, health endpoint)

Additional friction: re-adding the connector as a custom connector is blocked with "A server with this URL already exists", even after Disconnect, and there is no Remove option in the UI for this entry (see #73). So the stale registration cannot be cleared by the user.

This matches the symptom cluster in #552 (closed as not planned), #476 and #572, reproduced here with a directory/community connector backed by lemlist's official MCP server rather than a self-hosted one.

### What did you expect to happen?

A Connected connector's tools should be loaded into new conversations' toolsets, the same way first-party connectors (Make, Netlify, Google Drive) load on the same account.

### Steps to reproduce

1. Connect the lemlist community connector from the directory (OAuth, full API access granted)
2. Verify Settings shows "Connected" and lists all 58 tools
3. Start a new chat and ask Claude to list lemlist campaigns
4. Claude reports lemlist tools are not loaded in the conversation; only other connectors' tools are available

### Area

MCP Connector (adding/managing servers)

### MCP Server (if applicable)

_No response_

### Error messages or logs

```shell
"A server with this URL already exists." — shown when attempting to re-add the connector as a custom connector after Disconnect.

In-chat behavior (new conversation): model reports the lemlist connector is linked to the account but its tools are not active/loaded in the conversation, and lemlist does not appear among the available tools.
```

### Additional context

Screenshots attached: (1) connectors list with lemlist Connected, (2) Tool permissions page showing 58 tools, (3) MCP server manifest returned by the endpoint, (4) Add custom connector dialog producing the "already exists" error, (5) OAuth consent page (team name redacted).

Related: #552, #476, #572, #73, modelcontextprotocol/modelcontextprotocol#1675.

Note: unlike most reports in this cluster, this is a directory/community connector with lemlist's official server, not a self-hosted custom server, which narrows the possible causes on the server side.

Start by reviewing the related reports #552, #476, #572, and #73, then reproduce the lemlist directory connector flow described here. Compare the Connected state and 58 listed tools with a new conversation's available tools and the custom-connector "already exists" behavior. Done means the connector's tools load in new chats and the stale registration can be removed or re-added.

Scritto dal modello di indicizzazione a partire dal testo della issue.