FastStream Prometheus: use the default registry and make the metrics path opt-in #321

FastStream is the only bootstrapper that serves a private registry: FastStreamPrometheusInstrument builds a fresh CollectorRegistry unless prometheus_collector_registry is set (lite_bootstrap/bootstrappers/faststream_bootstrapper.py:188-203), while FastAPI, Litestar and FastMCP use prometheus_client.REGISTRY (docs/introduction/configuration.md:86-88). Collectors a FastStream service registers the usual way never reach its /metrics.

It is also the only one whose Prometheus instrument needs more than a path to be configured. prometheus_metrics_path defaults to "/metrics" (lite_bootstrap/instruments/prometheus_instrument.py:12), so is_configured additionally requires prometheus_middleware_cls or prometheus_collector_registry (faststream_bootstrapper.py:205-210). #229 listed defaulting the path to "" on FastStream and deleting that override as the cleanest rule, and set it aside only because it is breaking.

Work

• Test first, both failing today:
  • a FastStreamConfig with prometheus_middleware_cls and no prometheus_metrics_path skips the Prometheus instrument with the base reason ("prometheus_metrics_path is empty or not valid");
  • with a path set and no injected registry, the instrument's collector_registry is prometheus_client.REGISTRY.
• Override prometheus_metrics_path: str = "" on FastStreamConfig and delete the is_configured override, along with the extended not_configured_reason at :195-198.
• Default the registry to prometheus_client.REGISTRY and delete _make_collector_registry. Keep prometheus_collector_registry as the way to pass a private one.
• A second bootstrap in one process needs no reuse code, unlike FastMCP's tool-call collectors (lite_bootstrap/bootstrappers/fastmcp_bootstrapper.py:89-104): FastStream's MetricsContainer looks up existing collectors in the registry before creating them (faststream/prometheus/container.py:188-189 in 0.7.4), and two RedisPrometheusMiddleware(registry=REGISTRY) instances construct without error. Pin the multi-broker case from fix: instrument every FastStream broker, with per-type middleware classes #299 in a test.
• Docs: docs/introduction/configuration.md:70-88 and docs/integrations/faststream.md.

Verify: the new tests pass, and a FastStream app with prometheus_metrics_path="/metrics" and a broker middleware serves the broker metrics plus the default process and platform collectors.

Breaking for FastStream services that set only prometheus_middleware_cls: /metrics stops being mounted until they set prometheus_metrics_path. Services that scrape it see the default collectors and anything else on the global registry in addition to the broker metrics.