The answer to your question requires a lot of math but in my opinion, thinking about derivatives and integrals in terms of signal responses is not at all intuitive since your have to first interpret the signal in terms of its fourier transform (so a sum of different magnitude and frequency sinusoids) and then you have to think of the operations as phase shifts that are only applied to a certain set of frequencies (cosines and sines are 90 degrees out of phase and are the derivates and ontegrals of eachother). The integrator and differentiator will apply the phase shift to different sets of frequencies. What complicates things are that the magnitudes are also changed based on the frequency of the sinusoid. I think the person you are replying to happened to learn this stuff before actually doing calculus and to him it seemed intuitive. If you would like to learn more, any control systems textbook will be able to do the work a lot more justice than I can.

In simple terms, every sin(wt) or cos(wt) term becomes wcos(wt) and -wsin(wt) when differentiated, so the higher the frequency, the higher the multiplier.

This is not what most people would call high pass filter, as there's no cutoff frequency etc, and the phases get out of whack quickly, but you can think of it in these terms if it helps.

If you call multiplying the Fourier transform by a linear function "high-pass filtering" and dividing by a linear function "low-pass filtering", then I understand what you are saying. But I'm not accustomed to calling those high-pass filters and low-pass filters. I think of high- and low-pass filters as multiplying the Fourier transform by a step function (ideally, or some close approximation of such).