Limiting to console makes it kinda difficult. That Pretty much excludes nearly every MMO and most RTS games that could be considered competitive. Like Eve Online or New World. Games that have a major focus on pvp.

On console you're pretty much stuck with FPS games or brawl/fighting games.

I guess Age of Empires 2 is out now on Xbox lol. But RTS games on a controller don't feel very good.

Chivalry 2 can be played in 3rd person and can be as competitive as you make it. Nothing more satisfying than smacking someone with a two handed hammer and seeing them rag doll.