Do the compute instances not have hard disks? Because it seems like whoever's running these systems doesn't understand Linux or containers all that well.
If there's a hard disk on the compute nodes, then you just run the container from the remote image registry, and it downloads and extracts it temporarily to disk. No need for a network filesystem.
If the containerized apps want to then work on common/shared files, they can still do that. You just mount the network filesystem on the host, then volume-mount that into the container's runtime. Now the containerized apps can access the network filesystem.
This is standard practice in AWS ECS, where you can mount an EFS filesystem inside your running containers in ECS. (EFS is just NFS, and ECS is just a wrapper around Docker)