Here is the problem as I see it, Lafayette. I meet poor Americans every single day in my line of work. I am often surprised at their levels of honest, simple ignorance of how basic finances work. The biggest problem facing the poor is not the tax structure and self-dealing by the wealthy, and deference to their wishes by politicians and the wealth-retention industry. That is a problem in every society, whether we are talking about the United States, the People's Republic of China, or Europe. You are a Frenchman. French bankers own the commanding heights of your economy and your society's politicians just as much as American Bankers own ours. Rather, the problem the poor of our nation face is one of continually-perpetuated ignorance.
Without evidence to the contrary, I would argue that the already-wealthy's ability to pull ahead does not affect the poorest's ability to attain wealth, so long as they have knowledge of the tools needed to achieve wealth. I think the solution would be instilling awareness of these facts and in-depth financial education starting in high school to supplement already-existing basic economics courses. The United States has no generalized requirement for states and localities to mandate early financial education for children, which I think is insane for the wealthiest country on Earth. So impoverished children born into poor families whose parents have no financial literacy are expected to be taught by their ignorant parents.
With the exception of those children who manage to find financially astute mentors to guide them, or who are exceptional autodidacts, most will perpetuate the cycles of poverty they learned from their parents, such as relying on predatory payday lenders as creditors of first and last resort. If people were taught practical lessons showing how interest rates worked; what 401(k)/403(b)/457(b), or Individual Retirement Accounts were; how to budget; how to save; how to invest; how to pay your federal and state income taxes (and how income taxes worked); the importance of savings (both emergency savings and "sinking funds"); the pitfalls of debt (including student debt) and how to avoid it among other key pieces of financial knowledge, I think we would see a burst of wealth creation from within the poorer quarters.