I don't agree with your first point. Determinism has nothing to do with using random numbers to solve a mathematical problem approximately. In Monte Carlo integration you just sample values in a domain, and then add up the sampled values of the function being integrated.
However, I agree with your last point: you need computers to do integrals this way. Before computers, sampling methods were indeed used (eg gaussian quadrature), but the sample points were chosen very carefully to reduce their number, and reduce the work of human "computers".
However, I agree with your last point: you need computers to do integrals this way. Before computers, sampling methods were indeed used (eg gaussian quadrature), but the sample points were chosen very carefully to reduce their number, and reduce the work of human "computers".