H#LL No! An instructor should NOT have to be affiliated with a shop. I am an independent instructor, and I had to pay for every dime of every hour of my training on my own, without any help from anyone, especially an LDS.
When I approached my LDS about becoming an instructor, the manager told me I should not pursue it because it was too expensive, I would have to travel to get it, it was too hard, the insurance was too costly, it was too much responsibility, and it was just too much hassle to go through. In other words, he didn't need any more instructors, so he didn't want any more local competition.
I can be much more objective in helping my students. I am not obligated to get them to buy any specific brand simply because I'm told I have to. I have to work harder to get my own resources, but my classes are more flexible with scheduling, I can spend as much time as I'd like with students (no extra charge), I can set the number of students I want to teach at one time within maximum agency guidelines, and I don't have to follow any shop limitations, or deal with being managed by shop personalities. And yes, I am known locally for teaching the highest quality courses, with many referrals, so it's not a comparsion of the quality of education one gets from an LDS vs. an independent instructor. The quality depends solely on the instructor.
In short, if you pay for your education, regardless of what type, then you should be able to do what you want with it. It's no different than any other type of work in a free democracy.
With this being said, independent instructors and their LDS should try to work together to support each other. The shop can send their student overflow, or students with scheduling problems to the independent instructor, and the independent instructor can buy gear and air fills, and send students to the LDS to buy gear. It can be a local battle, but it should be a win-win for all.