You can't pay them significantly more, for certain court cases (especially for copyright violations) there are hard caps on a number of fees the attorney can raise. Plus you can get a significant sum of the cost paid by the state, even IF you loose.

The poor person is not "capped" by the law, but by their inability to pay more, which is the case in both systems.

In a way, but the downside to the losing party is limited. Also in (at least some of) these systems, it is not an automatic order for the losing party to pay, the court can decide to order it or not.

I don't think there really is any feasible way to prevent rich people and companies to have access to more/better representation.

No. They can't. That's the point.