Sounds cool in theory, but my small experience with a past company's serverless API is that it made impossible to debug. It was like deploying to a black box. There's no way to hit the API locally, etc.
I've seen that situation as well. My approach recently has been to build things where running on Lambda is a feature, and not so much an architectural decision. The aws-serverless-express library basically wraps an express (or other similar framework) app to handle interfacing with the lambda, marshaling lambda "event" data into the framework request interface, for example.
zappa (for example!) lets you produce a more or less normal flask app that you can run locally, or deploy to lambda / apigw with one command. another command let’s you stream the output of the code as it runs on lambda.
pretty sure similar tools exist for some of the other supported languages.
This is my experience. We will use Azure functions occasionally if all we need is to get data from a third party (with auth, when they can't set up CORS properly), but that's about it.
