Precisely this reason, because when things are run as root, they are owned by root and files it’ll create are root-owned, too. Typically–as is the best practices in server administration–only the most system-critical services should be run as root, because it’s the super user; it has the potential to bring down the entire system if it misbehaves or if exploited.
And internet-facing services, such as game servers, web servers, and its ilk are often very large projects which are impossible to keep exploit-free (or even just bug-free). Common services that do run as root are like SSH, which are so finely tuned over decades that it is very trusted to be root-run.
Anyway, all that ties in to “simply use a non-root user for Minecraft servers”. Both in the webui and in the terminal.
If, for example, you were root when you did rm -rf /, you’d trash the whole system. But as mc, you’d do pretty much no damage at all (to the system).
Likewise, the mc user has full ownership of the directories it needs access to, which are your server directories located within /var/games/minecraft/servers/servername. So running ServerStart.sh is 100% possible with your mc user, even if you can’t do anything to /var/ or /var/games/ or /var/games/minecraft/, etc.
To fix this, run the following commands as root (you have to be root to correct it):
chown -R mc:mc /var/games/minecraft/{servers,archive,backup}/servername
(replace server name with each server).
That’s for one server. You can shortcut it for all servers in your webui with:
chown -R mc:mc /var/games/minecraft/{servers,archive,backup}/*