Driving a loudspeaker
There's a key misconception in your question.

The electrical signal that represents a sound wave is proportional to the instantaneous pressure of the sound wave.

When a speaker is used to reproduce the sound, the pressure of the wave is NOT proportional to the position of the cone, but rather it is proportional to the cone's velocity. In other words, the coupling between speaker position and acoustic pressure acts as a high-pass filter (differentiator), and the low-pass characteristic (integration) of the speaker drive mechanism that you describe is exactly what you want in order to compensate for that.

It isn't perfect, but it works well enough. The rest can be taken care of by electronic equalization.