fix(private-server): run the MCP endpoint statelessly

[passcod]

🤖 Follow-up to #273 and #274. With Host and OAuth-discovery sorted, clients connect but then fail with Not Found: Session not found immediately after initialize — surfaced as "connected · tools fetch failed" / "Connection closed".

Cause

rmcp's default stateful mode keeps each MCP session in the handling process's memory and returns 404 for any later request whose Mcp-Session-Id it doesn't recognise. Behind the multi-replica deployment, the request after initialize is load-balanced to a different pod, which never saw that session → 404. Confirmed from Claude Code's MCP log:

Successfully connected ... capabilities: {"hasTools":true,...}
Error POSTing to endpoint: Not Found: Session not found
MCP session expired (server no longer recognizes session ID)
tools/list failed (MCP error -32000: Connection closed)

(A single curl reusing one HTTP/2 connection lands on one pod, so it worked — which is why this only showed up with a real client opening multiple connections.)

Fix

This is a read-only request/response API with no server-initiated push, so disable sessions:

• stateful_mode = false — each POST is self-contained; any replica can serve any request, no session affinity needed.
• json_response = true — return plain application/json per request, so there's no long-lived SSE stream for the ingress to buffer or drop either.

Tests updated for the stateless transport, plus a regression test that a tools/call succeeds with no prior initialize and no session id.