Update!
Wii U arrived and it works well. I've already tried WindWaker, Twilight Princess, Smash Bros, Mario 3D World and the Nintendo Land game it came with, and they all work perfectly.
FOR SOME REASON THOUGH, Breath of the Wild won't work. The disc appears perfect, but when I put it in the system, it makes some weird ticking sounds, takes longer than any of the other discs, and says "Invalid Disc"
Thankfully, I kept both the case and the receipt, and it hasn't been 90 days since the game came out, so I can exchange it.
Thankfully I don't need a Nintendo Network ID to play these games. Gotta wait for Nintendo to transfer my ID to this new console before I can get at least Mario Maker back. Hopefully I can get Mario Kart 8 back, which was pre-installed on the console. my 3DS e-shop shows the receipts for the DLC, so I'll surely get that back at least.