You are right and almost all methods of numerical integration (in any case all those that are useful and I am aware of) are equivalent to approximating the target function with another simpler function, which is then integrated using an exact formula.

So the estimation error is introduced at the step where a function is approximated with another function, which is usually chosen as either a polynomial or a polynomial spline (composed of straight line segments for the simplest trapezoidal integration), not at the actual integration.

Fortunately, for well-behaved functions, when they are approximated by a suitable simpler function, the errors that this approximation introduces in the values of function integrals are smaller than the errors in the interpolated values of the function, which are in turn smaller than the errors in the values estimated at some point for the derivative of the function (using the same approximating simpler function).