When I ride new horses, or new equipment, I always pester about the tack. ALWAYS. Especially when feeling out new people, and especially trainers.
I don't care. Personally, if I noticed the tack flaw, I'd just go up to the trainer and ask, "Is that a hackmore? Huh, wasn't sure. It looks weird, like it's missing a piece. Thought it was something new maybe."
I will talk to trainers in the middle of lessons, not to interrupt them teaching a lesson, but to inquire about what they are teaching and how the student is riding and progressing, and how the horse moves. I had a really good relationship with my instructor back in Memphis, and she knew of my mental blood thirst for horse knowledge, so she never minded if I stood with her and watched her teach. I actually like to do this too when feeling out new instructors. If I stumble upon them teaching a lesson, hell yeah I'm going to watch and ask questions. I need to decide if they're a good choice in trainer for me!
Heck, I tried taking lessons with someone new recently, because I moved to a new area. They had me riding in a Billy Cook saddle, it was a first lesson, just to feel out where I was as a rider before they could decide what to teach me, and the stirrups on the saddle were to short. I ride with a hellish long leg, even though the trainer who owns the saddle is about my height. I got off a bit into a lesson to adjust the stirrups. Got back on, found they still weren't long enough. The trainer said I was fine, my feet were where they were supposed to be. SUPPOSED TO BE?!?!?!
Darn it I was over breaking at the ankle and it frickin' hurt
Don't ride with that particular trainer any more.
I don't trust people, even "trainers" to know everything about what they are supposedly teaching.
Same goes with mechanics. Just because you can turn a wrench, does not make you qualified to work on cars.