Have been running our API on Gateway/Lambda for well over a year now, having migrated from EC2. We've spent almost no time "managing" API gateway, monitoring or logging since the initial setup and migration of our infrastructure.
If you are managing API gateway or your logging it's a failing on your initial setup and design, not on Serverless.
If you are managing API gateway or your logging it's a failing on your initial setup and design, not on Serverless.