Why are Chess, Mario, and Go not NP-complete?

I have a hole in my understanding of what makes a problem NP.

I understand that Mario, for example, is NP-hard - it can be reduced to the NP-complete problem of 3SAT (see https://www.youtube.com/watch?v=mr1FMrwi6Ew)

Therefore if these games were also in NP, they would be NP-complete (by definition).

My understanding is that NP problems can be solved in polynomial time by a nondeterministic turing machine that "guesses" each step correctly. Why isn't this the case for Mario, Chess, and Go? Can't such a machine just guess the right way, for example, for Mario to go at each gadget thus rendering the problem in NP (and thus NP-complete)?

It's a common misconception that chess is NP-hard. Generalized chess may be NP-hard. Chess has an 8x8 board, generalized chess has an nxn board with many pieces.

The question then becomes if generalized chess is NP-complete. I reason that it's not NP-complete; not because it's easier than NP-complete problems but because it's harder. So I'll reason it's outside NP:

Given a certain position on an nxn board, will white win if both players play perfectly? There may be a "yes" answer and the certificate for NP might be a list of perfect moves for both players, but it's intractable to check if those moves by black are actually perfect. Maybe black has better moves so that white doesn't win. You can't check such a certificate in polynomial time.

Also, it may even be the case that a game takes an exponential number of moves, so that such a certificate is of exponential length.

StackExchange Computer Science Q#70700, answer score: 11