> To freely paraphrase a famous quote (sorry I forgot who it's attributed to): it's not about strategy; it's about memorizing many famous games and game situations.
This is utter nonsense (probably that's why you "forgot who it's attributed to" - no one with any clue about chess would say so).
I'm just a fide master (two titles below grandmaster), but I feel more than confident enough to beat the average club or tournament player starting from a worse position if I happen to run into a line that they memorized from a book. Feel free to bring the book to the board, if you want to. It will not change the result (it might prolong the game a bit, because it'll help you to avoid some early mistakes).
There are two main differences between stronger and weaker players.
Especially at faster time controls, the difference in tactical ability (i.e. to "look ahead" a few moves by calculating the most likely continuations) is often more visible, stronger players are just faster, and more accurate (an ability that detoriates a bit with age, but we try to make up for it with experience, i.e. calculate "less", but the "right" lines instead).
Strategy: Stronger players instinctively "know" where their pieces want to be, and see a way how to get them there, or they recognize the weak points in the opposing position, and can quickly figure out how to adapt to that.