No, in general there is no such requirement (on Debian). Your ReadyNAS might do funny things (SELinux or whatever) making it difficult to do otherwise, but almost no systemd service needs to run as root.
203/EXEC means that systemd wasn’t able to launch anything. Does the exectuable
/usr/local/bin/syncthing
actually exist at that location?
Also, in another recent thread they said that ReadyNAS is actually a Debian derivative, meaning that it’s probably easier to install syncthing’s apt repository:
The repository will install the binaries and add a ready-to-use systemd service you just need to start & enable.