Name 1 year since 96 that he wasn't extremely relevant in the game..everyone you named besides wayne had short runs and then fell off, jay has consistently been at the top of the game, every couple years a new flash in the pan rapper comes along and then fades. Jay has lasted through all of those runs.
Also from 98-04 there's no question rap belonged to him, Dmx had his run but faded by his 3rd album, Jay sold 5 mil with Vol. 2, followed that up with the most successful tour in rap history that paved the way for rap acts to tour the way they do today, came right back with Vol. 3 and The Dynasty, and then completely shut it down with The Blueprint, he gave Nas an opportunity with that album/battle and Nas did what a great should do and stepped up and revived his career. There were more popular radio friendly acts that popped up during this run (Nelly, Ja Rule etc.) but the culture belonged to Jay/Rocafella.
He didn't have an unquestioned hottest newbie out rise like 50 but he also never fell off like 50 (live by, die by), Nelly had huge pop singles but no one ever considered him a serious rapper, these guys are in different categories, and Jay was the first one who could be compared to the street rappers and the pop acts and helped pave the way for dudes like Kanye who in turn paved the way for Wayne and Drake, to sum it up Jay had late 98 early 99, parts of 01 and 04 as undeniably the "hottest" rapper out and he is without question the most successful rapper ever followed by Em, who could've probably been more successful (not better) than Jay if he had the desire/greed to be