Yes, these containers are supposed to be stateless, but I was tasked with converting an app at my previous job over to using ECS on Fargate and we hit so many issues because of the limits on storage per container instance. We ended up having to tweak the heck out of nginx caching configurations and other processes that would generate any "on disk" files to get around the issues. Having EFS available would have made solving some of those problems so much easier.
I've also been wanting to use ECS on Fargate for running scheduled tasks with large files (50gb+) but it wasn't really possible given the previous 4gb limit on storage.