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Nintendo Switch Online doesn't pay for actual game servers unless the game was directly published by Nintendo. I shipped a multiplayer game on Switch and they don't provide a single service (other than the slowest and worst word filter I've ever seen that you are forced to use). I pay the server costs, and for a while my game didn't even need NSO to go online. It was like a year after the game was out they flagged it and forced the NSO requirement for online play.
Basically, someone clicks "Multiplayer", the game does a check to see if they have NSO, if true they can continue to actually play. That's literally it. It's just Nintendo getting their extra pennies from the player.