You will always have the wealthy with you.
There will always be those who do financially well within a particular economic system. A few will find ways to succeed no matter what the system. Others succeed only because they match up well with a particular system in a particular time. And as the system changes, or the person moves to a different society with a different system, their economic fortunes may rise or fall.
Degrees of economic inequality will rise and fall. Tweak or change the system and the inequality will rise or fall. But it never disappears. As long as wealth exists, there will always be wealthy people.
And even the population of who is poor and who is wealthy changes. Not all the poor stay poor, and not all those born wealthy stay wealthy.
None of this should be taken fatalistically, as though there's no point in trying to help the poor, as though we shouldn't make care for the poor a central goal of any civilized society. But neither can we avoid these realities. No set of policies will eliminate poverty. No government action will result in economic equality, or even near-equality.
The answer is not in what the government does. What it does will have an effect on society, of course. It can be more helpful or less helpful. The key, in my view, is on what is happening in the rest of society. You could have a critical mass among politicians and policy wonks that came up with the best policies a government could have to reduce poverty. But if there's no similar critical mass among the population as a whole, government action won't make a significant sent in the problem.
So go ahead, have wealthy people. But if they don't care about others, don't care about the poor, they won't be part of the real solution, and the problem will persist. And the government can create every conceivable safety net for the poor, craft every possible policy to give them a hand up instead of a handout. But if the poor are similarly self-oriented, and don't see a reason for self-improvement, you won't see a significant dent in solving poverty.
The poor and the wealthy you will always have with you. They are your neighbors. Love them. Show them the consistent, creative love of God, and (within limits) what the government does won't be that important.