Deep Health Check

Deep readiness check that exercises the SQLAlchemy pool.

Acquires a session and runs SELECT 1 with a hard timeout, retrying once on failure to ride out transient pool stalls or one-shot network blips. If the pool is wedged (phantom-checked-out slots) or RDS is unreachable across both attempts, the check fails and the ALB pulls the task out of rotation after its own unhealthy_threshold_count consecutive failures.

On final failure we return 503 directly via JSONResponse instead of raising HTTPException. The status to the ALB is identical, but the Sentry FastAPI integration only captures exceptions — so this path no longer triggers the "new unhandled error" PD rule on a transient blip. Sustained DB-unreachable is correctly surfaced by the ALB target-health alarm, not the Sentry new-issue rule.