xlink kai does work on stock consoles if the host is JTAG/RGH. The main trouble is the vast majority of games don't have system link support, and what support there is can be quite different to the baseline games.
There are two main models for multiplayer game servers
1) Simple multicast, be it central server or peer to peer (less expensive on bandwidth for the central server setup at the cost of dubious reliability and potentially anti cheat).
2) Central server that actually does something. Serves content, enforces rules, mitigates ping where it can (
https://www.gamedeveloper.com/desig...r-what-i-learned-coding-x-wing-vs-tie-fighter ), maybe calculates some things, holds data on world state (nobody wants to have to download and keep current the whole 100000 player base of a MMO, never mind all 100000).
You can usually work backwards to 1) easily enough. Figure out the handshakes, do some form of lobby/selection, fire data every which way it needs to go. This is what happened for the DS wifi resurrection projects, aided later by further exploits in the SSL/certificate handling.
2) Tends to be for things with leaked server code, server code shared with say PC versions of the game to allow self hosting of servers (rare in this day and age for... I guess control reasons) or simple enough that reverse engineering is possible. Stuff like the World of Warcraft private servers relies on some leaks, some serious efforts and the underlying protocol kind of being similar to a common database format. You don't necessarily have to replicate all of it if you can convince players to turn their ping down and just need positional stuff.
Having unhacked clients join can be tricky as said servers will often use encryption for various reasons -- enforcing bans, anti analysis (see something like pokemon battle analyser on the 3ds, though that was also Nintendo being legend class awful at network security), anti cheat/custom clients, money, control...
If you can break the encryption (unlikely -- I doubt MS did anything like the MD5 hashing we saw some people replicate certs for, and as they also own one of the more popular internet browsers in the world I will assume their handling is reasonably on form here) or work around it (unlikely on stock, the DS stuff being an anomaly really, doable enough if your hacks give you system level control).