On Discord, you cannot host your own server, and you cannot use any third party clients (without the threat of being banned).
You can host your own Matrix server, either on physical hardware, or a generic virtual machine you can rent from any number of ISPs. There are over a dozen compatible third-party clients (though many lack full feature coverage).
In summary, Discord is strictly a service. Matrix is a tool you can apply however you see fit.
It’s been a while since I used Arch, but it was smooth sailing while I did. In general, gaming means Steam, and Steam ships with its own runtime so it is not really impacted by whatever library versions are packaged by the distro. Gaming is a very common use case. You’d have to pick a pretty obscure one to find something where it isn’t tested and somewhat streamlined.