This is an excellent post.
I think the competition idea works at the level of a country if people by and large play by the rules (a big if).
It does not work at the level of macro economics.
If you (as a country) insist on competition and absence of mega corps in your country while a country next to you subsidizes mega corps that outcompete your smaller companies - you will lose.
Thus in a world of competition, every country will strive to get nuclear weapons to be able to compete militarily and then given that we're human, we will have conflict sooner or later and it will result in nuclear holocaust.
If you think fair enough but nuclear holocaust is just the price we pay and it's worth it to continue doing competition, then alright.
I don't think it's worth it, so there needs to be a fundamental re-think how we approach these problems.
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Regarding wealth inequality specifically - I agree with Aristotle who made the distinction of Aristocracy vs Oligarchy. Aristocracy being the rule of the few for the common interest and Oligarchy being rule of the few for the interest of the rulers.
Somebody will rule, it's the intention and overall direction that matters. I'm not sure that just re-distributing wealth addresses the problem - it does strike me as something an Aristocracy would surely do if one were to appear, so that much is well and good.