I agree with you. The rich are unlikely to make it so that there are no poor (after all, it is the lack of money that makes the poor work for the rich). And the poor, if he suddenly becomes rich, will be forced to accept the rules of the game, and will himself do everything to keep the poor. Yes, he will do charity, but it will be within the system, with the expectation that the number of poor people will not decrease (and will not increase much).
Indeed, it is undeniable that correlation exists between poverty and wealth within an economic system. However there are numerous wealthy individuals who actively engage in social works and make substantial contribution for humanitarian causes all over the world. An exemplary figure is Bill Gates who has dedicated significant portion of his wealth to supporting those who desperately need it.
In summary, both the poor and the wealthy are integral part of our society, we need to make efforts to create an environment, that safeguards interests of both groups..