What complexity class would this version of generalized chess fall?

Question

By now I understand that generalized chess is harder than NP, and is EXPTIME-complete for the decision problem "Given an nxn board with a given position, can white force a win?" because the proof would require an exponential amount of steps to show that each branch of the tree eventually leads to a win. Therefore it's not in NP.

But if you were to ask the decision problem "Given an nxn board with a given position, can white win?" and apply it to generalized chess, would this variation be in NP? Because the verifiability of this proof would only require polynomial time (just play out the winning game). For this decision problem, would it fall in NP, because a problem solvable by a DTM in EXPTIME and verifiable in polynomial time would be NP-complete, yes?