Assuming at the outset that he has it narrowed down to two numbers is incorrect in the sense that that requires its own proof, which I think it what is confusing people. It turns out, based on the parameters, that there are no possible groupings that would allow the interrogator certainty based on his last question larger than two, and there is only one such grouping. The video is not wrong, but it kind of skips steps in a way. It makes intuitive sense to assume that the answer is a perfect cube because that narrows down the options significantly, but that is a shortcut, and is not technically proven until you do the math.
I think what ultimately confuses people about all of these types of puzzles is that the solution is always presented as a combination of hard math and fuzzier seeming leaps that actually have hard math behind them. I think either using less actual math, or more actual math, would make these more clear depending on an individuals inclination and the nature of the particular puzzle.