It costs a lot to be poor. Even middle-class people pay higher interest on loans and receive less on savings. Poor people have limited access to internet resources, fewer (or no) social connections for opportunity, etc.
As Bernstein notes, even if tax policy does not yield significant redistribution, it can change the pre-tax choices of wealth. Tax rates that devalue windfall wealth and encourage long-term growth instruments and, more importantly, steady-growth company policy instead of boosting share-value through buy-backs and such, would be a start. Discouraging market concentration via mergers through tax policy might be easier than anti-trust regulation. A value-added tax would serve as a tariff, which could restore some trade balance.
We are not a capitalist society, or at least were not particularly so at the start. We are, however, a contract society as opposed to an ethnic identity or a shared covenant. We can renegotiate the contract every election..