Does't the proposed solution both solve the issue of FaaS and also remove the main benefit? (ie having stateless function with clean state when it starts runnin

Does't the proposed solution both solve the issue of FaaS and also remove the main benefit? (ie having stateless function with clean state when it starts running) Also it feels to me like FaaS is good when someone else is running the infrastructure (so you really don't have to know about the underlying infrastructure). Why are people self hosting this? Is it for local test platforms before deploying to a managed FaaS?