A key that lives in the central config is only worth centralizing if
publishing it does not cost a fleet restart. The backend bakes host and
Authorization header into its clients, so it now accepts a fresh
LLMConfig: clients are dropped and rebuilt on the next call, health
stays keyed by target name, and the probe follows the new hosts too.

The overlay's llm section is no longer always staged. It is re-parsed
against the local document and swapped live; only a backend without the
swap method or a document that fails to parse falls back to
restart_required.