Why are most people born poor or middle class while only a tiny percentage are born wealthy?

Karma and financial wealth have nothing to do with each other. Karma is acquired accordance with supernatural laws as a result of volitional (intentional) actions. Virtue leads to good karma, sin to bad karma.

Money on the other hand, is acquired by being succesful in an artificial man-made economic system, according to artificial man-made laws that reward skill, cleverness, shrewdness, etc. This has nothing to do with virtue.

Financial success and karma are therefore usually unrelated, and sometimes even opposed.

What could Jeff Bezos, Elon Musk, Bill Gates or even trust fund children like Paris Hilton karmically do to deserve that lifestyle while the vast majority of the world struggles?

Nothing. These people don't have good karma. They have skills or attributes that the economic system rewards highly, that's all.

We don't live in a fair world that reflects karma. Quite the opposite: the modern world is entirely artificial and goes against karmic laws. The only thing that is unavoidably governed by karmic laws is rebirth, so once parentage is still determined by karma. (But the forces of Evil are working on a very clever way of sabotaging even that karmic law.)

/r/Buddhism Thread