I didn't see the video mentioned in the OP since it's been removed, but this is what Kip Thorne wrote in the book: "Time, he realized [referred to Einstein], must be warped by the masses of heavy bodies such as Earth or a black hole, and that warping is responsible for gravity.", and again: "The greater the slowing of time, the stronger gravity's pull.".
I think that, since space and time are linked together, to make the explanation cleaner he simply uses the slowing of time to indicate space-time curvature. Don't forget that "The science of Interstellar" is mainly a divulgative book and indeed after these statements he continues his exposition talking about how the slowing of time is important in Interstellar.