I recently learned of a fork of ioquake3 (open source version of Quake 3) called Lilium Arena Classic, which is compatible with Quake 3 1.16n, unlike ioquake3. This means it is able to join the same servers as Dreamcast players, which was otherwise only possible with the old closed-source version.
I ran it as a client and was able to connect to the same servers as on my Dreamcast with no problems. I can't get a server hosted with it to work, but I don't know for sure if that's a limitation of it or if I've configured something wrong.
It currently only supports x86 processors so it can't be ported to phones or Switch or similar yet, but maybe someone could make an Xbox port of it, as there's a homebrew Xbox port of another version of Quake 3.