I am sure the training folks are truthful in their reports on the frequent failures of aftermarket triggers. But, like anything, common sense rules. I am convinced the most reliable match-quality trigger is a good two stage setup. John Holligers' tuned RRA triggers are great. And, for any target use, it is my preference. It has no adjustments, and mine have been totally reliable so far.
I think a good single stage trigger is better for tactical use. But, in my opinion, most triggers are unacceptable as delivered from the factory. Even if you smooth the bearing surfaces, the engagement angle is still a problem. From all reports, I am sure the Accuracy Speaks trigger is excellent. I am satisfied with the JP hammer/trigger combination. But, I removed the over-travel adjustment screw, used the full power hammer spring, set up the engagement adjustment for a deeper engagement than would be desired in a target rifle, and then used loctite to keep it there. I believe it would be reliable through the most rigorous training, but I can't say I have proven that. The real problem with single stage triggers in the AR is the safety design. If it had been designed to block the hammer, then the sear engagement would not be such a critical issue. But, since it only blocks trigger movement, there has to be enough engagement to avoid dropping the hammer from an impact.