refactor(db): use shared pooled DatabaseClient singleton instead of per-call instances

- Replace get_db_client() creating new DatabaseClient on every call with a
  module-level singleton initialized once at startup via init_db_client()
- Add init_db_client() and close_db_client() lifecycle functions called
  from FastAPI lifespan handler
- Migrate all DatabaseClient methods from legacy self.connect()/self.conn
  to pooled self.get_conn() context manager for thread-safe connection reuse
- Pool is properly torn down on application shutdown

Closes leeworks-agents/SPARC#7

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

SPARC/api.py

@@ -21,11 +21,13 @@ from SPARC.auth import (
     TokenResponse,
     UserResponse,
     check_jwt_secret,
+    close_db_client,
     create_tokens,
     decode_token,
     get_current_admin,
     get_current_user,
     get_db_client,
+    init_db_client,
 )
 from SPARC.types import BatchAnalysisResult, CompanyAnalysisResult
 
@@ -155,6 +157,7 @@ async def lifespan(app: FastAPI):
     """Initialize resources on startup, clean up on shutdown."""
     global _analyzer
     check_jwt_secret()
+    init_db_client()
     _analyzer = CompanyAnalyzer()
     # Mark any jobs that were running/pending before the restart as failed
     from SPARC.database import DatabaseClient
@@ -167,8 +170,9 @@ async def lifespan(app: FastAPI):
         logging.getLogger(__name__).warning("Marked %d stale jobs as failed on startup", stale)
     _db.close()
     yield
-    # Cleanup if needed
+    # Cleanup
     _analyzer = None
+    close_db_client()
 
 
 app = FastAPI(