what is actually happening to cause the interference
Current flowing in a conductor when it has a fast rise time radiates radio frequency energy. As a current pulse has an infinite number of harmonics it radiates a broad band interference signal.
can't change the wiring
So you are probably stuffed.
However, you can try and cut down the interference by reducing the rise time in the motors by using capacitors close up to the motor contacts. You could change the motor type so that it does not have a commutator circuit breaker or you could attempt to minimise the disruptive effect of the interference in your video circuits by increasing supply decoupling. The "larger supply" solution works because it is a lower impedance supply and so less prone to picking up the signal.
However curing the problem at source is often a much better strategy.