Old Nemesis was the first monster in the series that could follow you through loading screens. Came also using weapons, pronouncing further that it's capable of using powers that the player has considered so far to be exclusive to themselves.

It's not the fact that it pursues that makes it memorable, but the fact that it rivals the player. That's why Nemesis feels less akin to Mr. X, that ultimately is just another "dumb" monster that chases you around to fist your ass. Mr. X has some interesting elements with the hat, as it kinda implies that its capable of some further thought than the random zombies, but ultimately it does nothing particularly rival-like. The only time I remember something similar in a modern game was in Nier Automata when you fight the gay machine brothers and they start leveling up. I think the only way to give the same impact in modern RE format would be to make pursuer player-controlled souls styled invader that knows how to counter the player's tricks.

Actually, that reminds me that mr. X did actually start holding hand in front of its face if you kept headshotting it? That was actually pretty neat.