Do you think it would be possible to create a solution that didn't infringe copyright?
Technically if we made a private server for say Quake 2, that would be fine. It's only if we made one for say Splatoon 2 that would land someone into hot water.
Now if something like Splatoon 2 were to offer the option to host your own private server from your own computer, that would be a different story.
This gets into some really muddy issue with copyright laws, but basically it boils down to what the company permits you to do with their IP. It would have to be on Nintendo (or any other company) to give you the OK and everything needed to host your own server with their IP.
Of course there's also the case where servers are hosted after the system has lost support and those cases don't really seem to be touched by most companies. It seems like Nintendo hasn't really done anything about Wiimmfi. Which is most likely because the courts would be far more inclined to be in favor of Wiimmfi over Nintendo's claims. Of course this is just a theory on my end, but one based on my knowledge of Nintendo's history only taking on battles that they know they can win.
Edit: I forgot to add that any private server being used to bypass their paid online service is against the law. So there's that issue that needs to be taken into consideration.