You asked everyone's opinion. If I was in your situation and the trainer asked me to carry a crop, I would respect their request whether I would have to use it or not. In the large scheme of things, it really doesn't matter. I wish the biggest thing I had to worry about on any given day was whether or not I should carry a crop.
Also, not to sound full of myself, but I (or any respectable trainer that I know) can read a horse well. If I (or again other trainers) were working with a student 3 days per week for a month, I would be pretty comfortable in how well I know your horse. I usually do a one hour consultation with people and already know their horse better than they do. Other professionals that I know (trainers, farriers, dentists, vets, etc) can read horses just as well or better.
No one is saying beat the horse with a crop, they are just saying to respect the opinion of the person that you are paying or find someone whose opinion you agree with. Yes, it is often the riders fault that the horse refuses, just because you are carrying a crop doesn't mean that you have to beat her over the jump if you are messing up. To be honest, if you are that worried about responding like that, then the little kid jumps is where you should be until you are more secure in yourself and in your horse. If you don't want opinions, don't ask the question.