Should a very competitive person pursue a career in academia after getting a PhD?
I don't see the problem. You are hardly alone in having the qualities you described -- academia is full of very competitive people, and competitiveness is also the reason why many of those people are highly driven to work hard and excel at what they do, and therefore to some extent the reason why it is such a fun and interesting place.

I understand that you are worried that being overly competitive will make you perpetually frustrated and unhappy, since no matter howsuccessful you become there will always be people who are yet more successful than you. To the extent that that's true, it will also be true outside of academia in any environment you are likely to end up in as a talented and highly educated person, unless your alternative plan is to give up all ambitions and pursue a lackluster life managing a bagel store or something similar. And honestly, given your description of yourself, I really don't see you doing that. So you're stuck with your ambitions, like it or not, and academia is as good a place as any to be a person with such traits.