we seem to use rms even when we don't square anything and we aren't interested in calculating power...

Yes, we do. We calculate the current through a capacitor by using the RMS quantities - there is no net power dissipated or taken because the capacitor is purely reactive yet, we still use RMS terms. If the voltage is a sinewave at 100 V RMS and the capacitive reactance were 100 ohms, then the RMS current is 1 amp.

• If we used some other measure of voltage (say) Vpk then we would get Ipk.
• If we used the average of a sinewave the result would be zero

What does mean-root-square mean? In effect squaring then rooting a sinewave is like perfect rectification of the sinewave. Then, if you take the average it is 0.63689 of Vpk. But feeding this "corrupted and harmonic rich" signal into a capacitor is nonsense if you are expecting calculate current related to a sinewave across its terminals.

we seem to use rms even when we don't square anything and we aren't interested in calculating power...

Yes, we do and, we also calculate the current through a capacitor by using the RMS quantities despite there being no net power dissipated or taken (the capacitor is purely reactive) yet, we still use RMS terms. If the voltage is a sinewave at 100 V RMS and the capacitive reactance were 100 ohms, then the RMS current is 1 amp.

• If we used some other measure of voltage (say) Vpk then we would get Ipk.
• If we used the average of a sinewave the result would be zero

What does mean-root-square mean? In effect squaring then rooting a sinewave is like perfect rectification of the sinewave. Then, if you take the mean (to get mean-root-square) it is 0.63689 of Vpk. But this is meaningless to a capacitor and even if we ignored the "mean" part, feeding this "corrupted and harmonic rich" signal into a capacitor is nonsense.