Right that was my point. You most likely have to load the whole thing into memory, but in the rare case that you can seek directly to the right spot in the file, ECS might have an advantage for you in Lambda.
EFS/NFS makes a lot of sense I think when you have a lot of random/unpredictable access across large numbers of files. You could do the same thing with FUSE/S3 but across lots of smaller files/accesses I would think the overhead adds up faster.