How is that any different than a typical web server? Hopefully you don’t maintain state on your web server either. This isn’t a new concept. Even in the early 2000s when I was maintaining two web servers behind a physical load balancer we knew to maintain session state separately from the web server.
I hate to be “that guy”. But “the cloud” is no more than someone else managing servers and services at varying levels of abstraction.
I have no idea why some people hear AWS/Azure and they act as if the same architectural principles that have always applied still apply, just on a more granular scale in the case of “Serverless”.
More of a rant about the person who complains about not being able to maintain state at a tier that you never maintain state at in even a traditional architecture.