Forget about narratives and actually watch him in his Liverpool days. His movement off the ball was always incredible, always managed create chances for himself, he's tactically intelligent, and his technique was always top notch. Yes he improved his finishing under Pep, but he never needed to learn how to pass, control the ball, dribble in tight spaces, and play off his teammates when Pep arrived - those are skillsets he had by the time he signed his first professional contract.
Yes managers can improve players, but they don't magically transform players into completely different types. The best manager in the world can't coach Adama Traore to have B.Silva's football intelligence, just like they can't magically make Dembele a Barcelona quality passer at the age of 22.