Okay, after looking at the logs on the docker container as I was trying to run it, I found the following:
ERROR: unable to configure virtual network port: could not open TUN/TAP device: No such file or directory
Lots of rabbit holes lead me to here ( Running zerotier-one on synology nas using docker ) where I saw that they change the permissions on the /dev/net/tun so I did that and it worked.
So, ultimately it seems to come down to following the instructions here ( Synology | ZeroTier Documentation ) except for changing the permissions on the /dev/net/tun:
chmod 0666 /dev/net/tun
And using the following docker run command:
docker run -d \
--name zt \
--restart=always \
--device=/dev/net/tun \
--net=host \
--cap-add=NET_ADMIN \
--cap-add=SYS_ADMIN \
-v /var/lib/zerotier-one:/var/lib/zerotier-one zerotier/zerotier:latest