Alright, I finally got my Tailscale Docker container to work with help from this Reddit thread.
It looks like the Tailscale Container ignores everything that's put in the "Environment" section of the Docker Compose file. In order to get the container configured properly, I had to do it from the SSH shell of the OpenMediaVault host.
First I had to pull then start the Tailscale container by clicking the "Up" button in the Docker Compose Files menu. Checking the container logs, that's where it failed to connect to the Tailscale network with the error:
tailscale | 2024/06/03 10:48:11 Switching ipn state NoState -> NeedsLogin (WantRunning=false, nm=false)
Then I logged in to the OpenMediaVault shell throuigh SSH. From there, I executed the following commands (assuming the name of the Tailscale container is "tailscaled"):
docker exec tailscaled tailscale status returned "logged out". This confirmed that the container was running but couldn't log in to the Tailscale network.
Then I ran the following command:
docker exec tailscaled tailscale up returned an URL where I went from my web browser and allowed me to log on the new machine into my Tailnet. If you want to use a key instead, I suppose using the command line "docker exec tailscaled tailscale up --authkey tskey-abcdef1432341818" would work but I wasn't able to test that since I used the commands that the Reddit thread referenced.
After logging on the machine into my Tailnet, it appeared in my Tailscale Admin Centre. Now I needed to advertise it as an exit node and publish a local route, so I had to run the following commands to achieve that:
docker exec tailscaled tailscale up --advertise-routes=192.168.1.0/24
then
docker exec tailscaled tailscale up --advertise-exit-node
Once all of these had been executed, the node showed up properly configured in the Tailscale Admin Centre and I was able to use it to its full potential. I can also shut down the container and restart it and it retains the correct configuration.
Hopefully this helps someone else at some point.