The strict policy forbidding cross-training at any time is likely a sign that the instruction you are receiving from your current school is the best around, and your coach simply doesn’t want your progression to be sullied by the inferior techniques on offer at other schools. As a paying member of the gym, you owe your complete and undying fealty to the gym and its staff, and violating this bond by taking a class with a friend or partner at their school, or checking out lunch classes near your job, would be nothing shy of the ultimate betrayal, and your coach’s insistence on this just shows how much they care about you and want every class you take to be of the utmost quality. If you weren’t receiving the absolute peak experience anyone could ever ask for in a martial arts school, why else would they insist you only train with them? There’s literally no other explanation than your coach knowing that if you trained at new school for a class you would definitely find the experience to be unbearably worse and hate your time, so they’re protecting you from wasting an hour of your life.

Very Secure Coach Probably Only Forbids Cross Training Because Your School is So Good
That’s gotta be why.