Trouble mounting EFS Access Point to ECS Volume
Is it possible that you have a mismatch between the defined uid/gid in the Docker image you're trying to run and the 1001/1001 uid/gid you are attempting to use?
We ran into this same error when we took a base image, which had volumes defined, and tried to alter the primary user's uid in the image. Since you can't alter file ownership or permissions of a volume, once its been created, the docker daemon would attempt to apply the original uid ownership to the mounted EFS volume, causing the EFS access point to block the change in ownership/permissions.
You can check this by shelling into the docker container locally or running a docker exec to ls -la /path/to/container/folder and check if the ownership of that folder is set to your user with uid 1001.