In this way we could avoid having to use public servers, where our jams could be invaded by strangers, and also avoid slowdowns due to heavy traffic on public servers.
Bear in mind that there are other complications with running your own server.
You'll either need a fixed IP address
or you'll need to rely on some sort of public discovery service provided by the app developers (which might have similar privacy issues to public servers).
You may have to make changes to the firewall settings on your Mac and/or your home router (and if you've got the home router that your ISP provided it may well have had the more advanced features nobbled). Making sure that you're not exposing your home server to security breaches will be your problem.
All the data flowing to/from you and your fellow jammers will have to go through the bottleneck of your home broadband connection - and even "fast" broadband is often optimised for "media consumption" meaning that the
upload speed is only a fraction of the download speed - and your server will rely on that upload speed to stream everything to
each of your friends. Slow internet may be down to overloaded public servers, but can also be caused by your neighbours all bingeing box sets.
I'd never discourage anybody from "having a go" but generally I'd say that the best bet for anything that is going to be shared outside your household is to either stick to public servers, or, (if you want to take on the technical challenge and aren't afraid of a spot of linux) get a virtual server "in the cloud" which, at least, should have a fixed address and far faster connections to the Internet than your home (and, if it gets hacked, just delete it).
That doesn't apply to using Home Sharing or Plex as a media server within your house, of course.
...and, no, the only unique features in Mac OS Server last time I looked were things like managing multiple Macs/users across an organisation and hosting networked Time Machine backups (and even that can be done DIY with a bit of googling). You don't need it for running third party server apps or standard Internet services like web servers.