Why do we monitor on moving coil loudspeakers?
Strictly speaking, the loudspeaker is the entire thing consisting of drive units, crossover and enclosure (cabinet). So here I will refer to the moving coil drive unit, which is a component of virtually ever loudspeaker you will ever hear.
The moving coil drive unit consists of a magnet, a coil of wire which when the signal is applied to it becomes an electromagnet, and a cone-shaped diaphragm that pushes against the air creating sound waves.
Moving coil drive units are simple, efficient and generally work quite well. The drawback however is that the 'motor' - the magnet and the coil - is in the center of the diaphragm, and the diaphragm must be stiff so that vibrations travel quickly outwards from the center to the edges. This must happen without the diaphragm bending, otherwise the center could be moving outwards while the edges are still moving inwards as a result of the previous half-cycle of the waveform. This effect is known as 'break up' because the diaphragm is not moving as a whole but various sections are moving independently of each other.
In the low frequency drive unit - the woofer - this causes audible distortion, which is greater at higher frequencies. It is a fact that distortion of this kind always exists in a moving coil drive unit. There has to be a compromise between the diaphragm being stiff and being light. Obviously if the diaphragm were made from 10 mm steel it would be perfectly rigid, but it would also be heavy and the drive unit would be inefficient.
Another kind of drive unit - the electrostatic - does not suffer from break up at all, hence distortion can be almost entirely absent. The flat diaphragm of an electrostatic drive unit is driven over its entire surface area, not just from the center, hence there is no need for it to be stiff - there is no way it can bend.
The trouble is that electrostatic drive units are expensive and inefficient, therefore electrostatic loudspeakers are sold only in very small quantities.
So even though an electrostatic loudspeaker, used as a studio monitor, would tell you the absolute truth about what's on your recording, it isn't representative of the real world. So we monitor on moving coil loudspeakers simply because they tell us what it's going to sound like for the listener.