Would you support it if a car towing a caravan with defective brakes jack knifed and crossed the carriageway causing a head on collision that killed someone close to you?
Anyone, me included, would probably become a campaigner if personally affected by something of this sort.
But introducing an MOT test for trailers does not guarantee that accidents like this won't happen (because the trailer might not have a valid test cert, or the fault might develop between tests). In any case, the most common trailer accidents are by jack-knifing following out of control snaking and I don't see much that an annual MOT would do to reduce the chance of that happening. As a control measure for mitigating the risk, the cost and bureaucracy burden is out of all proportion to the likelihood of occurrence.
When we pulled up at reception on one of the CAMC sites in Scotland last year, accompanied by our typically lightly squealing brakes, the Warden came out of the office to great us with 'as I was just commenting to my colleague, here comes another Alko chassis'.
Nothwithstanding all the theoretical, terrible and rare issues there might be with the braking system I think it's reasonable to share with the OP the practical knowledge that lightly squealing brakes are very common.