My guess is that you still need a Nintendo server to run the game. Otherwise, you would need to have extra stuff added to the update to generate the new "server" on the LAN mode. Doing this means that one player is going to be both host server and client, which means extra CPU is need to do both task. Having 8 players running around spreading ink might be CPU intensive and hurt whoever is playing the server/client machine. From a technical standpoint, I've seen matches where so much ink is flying around it causes game lag (guess because so much CPU processing is needed to figure out all the overlapping ink splats). From a technical standpoint, I've done a lot of game design over the past decades. So I can see this game using a lot of CPU because it has to keep track of more than just which players are running around and shooting at each other. It has to calculate splat volumes, splat distance, ink arc, speeds, and keep track of which areas are painted (walls, floors, objects, etc.)
I think it serves another purpose. If the next update gave you a copy of the "server" code then it opens it up to looking for hacks or exploits to use against the official Nintendo servers.