Explicitly initialize Stats in API server lifespan

[jscheffl]

Re-apply #68078 by @diogosilva30 with fixes after a bad review, merge and reversion in #68481:

Fixes from the original PR:

• Not mentioning "Task SDK" anymore as this is only about initializing stats correctly in API server
• Remove caplog as we actually wanted to ban this
• Double checked: no newsfragment. really.

Why I am re-applying this PR: Because in all Airflow components the "Stats" is correctly initialized, but missing in API server.

closes: #68077

Original PR quote:

Why

The API server serves the Edge Worker REST API (/edge_worker/v1/...). A worker heartbeat (PATCH /edge_worker/v1/worker/) runs set_state → set_metrics, which records edge_worker.* metrics through the Task SDK Stats singleton (resolved by the Edge provider via airflow.providers.common.compat).

On main: hardening against a fragile import side effect

The bug does not reproduce on current main. On main, task_runner.py imports serde at module level, which triggers serde._register() → Stats.initialize(...) as a side effect. That fires on any import airflow — including in the API server — so Stats gets initialized accidentally.

However, this is fragile: it would silently regress if the module-level serde import in task_runner were ever made lazy, or if the mask-secrets import chain changed. The API server doesn't actually depend on serde, so relying on this side effect is not intentional.

Adding an explicit Stats.initialize(...) in the FastAPI lifespan makes the dependency intentional and self-contained. It's a no-op on main (already initialized) and removes the accidental coupling.

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