The official Telegram MCP from Anthropic is vibe-coded garbage. You can just use /mcp, select Telegram and choose Reconnect and it instantly works. But it disconnects when you least expect it.

The solution is to fork the plugin, fix it yourself, and use your fixed version. Maybe rewriting it to use Webhooks it the better approach.
The trade off is you have to use the poorly named dangerous channel flag and it requires confirming that you intend to use the flag each time Claude opens.