Hmm, well it doesn't seem to be any problem with the docker compose then as best as I can tell. I picked a random ext4 flash drive and replicated your setup with the UID and GID set and it seems to work fine:

# /etc/fstab
/dev/sda1       /home/<me>/mount/ext_hdd_01  ext4    defaults 0 2

~/mount % ls -an
total 12
drwxr-xr-x  3 1000 1000 4096 Mar 27 16:22 .
drwx------ 86 1000 1000 4096 Mar 27 16:31 ..
drwxrwxrwx  3    0    0 4096 Mar 27 16:26 ext_hdd_01

~/mount/ext_hdd_01 % ls -an
total 6521728
drwxrwxrwx 3    0    0       4096 Mar 27 16:26 .
drwxr-xr-x 3 1000 1000       4096 Mar 27 16:22 ..
-rw-r--r-- 1 1000 1000 6678214224 May  5  2024 PXL_20240504_233345242.mp4
drwxrwxrwx 2    0    0      16384 May  5  2024 lost+found
-rwxr--r-- 1 1000 1000          5 Mar 27 16:27 test.txt

# ~/samba/docker-compose.yml
services:
  samba:
    image: dockurr/samba
    container_name: samba
    environment:
      NAME: "Data"
      USER: "user"
      PASS: "pass"
      UID: "1000"
      GID: "1000"
    ports:
      - 445:445
    volumes:
      - /home/<me>/mount:/storage
    restart: always

I was able to play the PXL.mp4 video from my desktop and write back the test.txt file

Have you checked the logs with docker logs -f samba to see if there's anything there?

Also you could try to access the HD from within the container, using docker exec -it samba bash and then cd into /storage and see what happens.