Do they run the kennel club good citizen scheme classes at all?? If they do it may explain the no headcollars/harnesses rule as under -*test*-('") conditions they cannot be used.
I do allow headcollars/haltis and harnesses initially, but we always work on reomoving them in class so eventually the dog is able to walk without the use of these training aids. I would not however tolerate the use of choke chains in my class - just my rules.
People outside of my classes still use headcollars from time to time for safety reasons and if walking more than one dog, or if there is a long lead walk before off lead play and they are time restricted with training loose lead work on a flat collar.
You could persevere with the flat collar in their classes, and use the headcollar outside, you may well find that you may end up not needing the headcollar in the long run.