Lambda is cheap, API Gateway is expensive.
Lambda with API Gateway should be used for low or unpredictable volumes.
You can replace API Gateway by an ALB, which will be much much cheaper in the article use case.
Lambda is also great when it's triggered by everything but API Gateway. Analysing a file uploaded on S3, reacting to an SNS message or consuming messages in SQS.
It's self-hosted.
The website seems down at the moment though.