In my opinion the working of a CMC is to stop RF-currents on the outside of the braid, whether it originates from the antenna or from surrounding mostly vertically polarized disturbances. Thus these currents can not reach, or at least will be mitigated, the receiver. And the "return" current from the antenna will be confined inside the coax, so I don't have both a vertical antenna (if the coax is mostly vertical up to the antenna) by the coax and the real antenna. Because of RF skin effect the inside of the coax braid does not know anything about the outside and vice versa (if proper coax). I have made a simple instrument to measure the current on the coax outside, see picture. Without an CMC I can get high readings from currents on the outside, with an CMC there is no indication at all.
73/Torbjorn