I've been running a Matrix server for a half a year and I have not run into any problems. The main thing that surprised me was how much database space Synapse took on my server but that was an easy enough issue to deal with. That may also have to do with the fact that I'm in some of the most complex rooms that exist on Matrix.

I tried running a XMPP server for a bit but that was a rather frustrating experience between getting it to integrate well with the rest of my infrastructure, figuring out what configuration was needed and what XEPs were needed, what clients had the features I needed, what clients supported the OSes I used, and spam. To be fair, the reason I thought setting up Synapse was extremely easy may have been because I did it with almost twice as much experience with devops than I had when I tried to setup the XMPP server.

Regarding registering with a central authority, you're probably thinking of identity servers. Identity servers are used to provide 3PID (email/phone number) -> Matrix ID mappings and are completely optional. There have been attempts to decentralize that as well but for one reason or another, no good way to do it has been found. At least the source code of the identity server is open source[1] so there is that.

Otherwise, to set up a Matrix server, you simply have to either set up a json response from "/.well-known/matrix/server" or an SRV record on DNS that points to the Matrix server.

[1]: https://github.com/matrix-org/sydent