Your trainer is right to believe your dog "should" learn to be obedient on a flat collar. However, he is clearly not doing what is needed to get you to that point. It is very difficult in a group class to handle people using different equipment, as he would have to give individual instructions. It can be done, but is in fact rare to find a trainer who can swing it.
Shoulder injuries, back injuries, and people just too small to physically control their dog, however, are extremely common, and it's too bad he hasn't been handling the situation better.
As for your dog, leash aggression, again, very common problem. Some dogs grow out of it or are trained out of it; for others it is a life-long issue, but it does become manageable. Your instructor may or may not be willing to let you deal with the problem in your own way.
Frank and Chloe will be displaying very dominant signals to each other, ears up, tail up, lips forward, chest forward, maybe hackles. In order for these signals to take full affect, your dog needs to be standing. Lying down is a submissive posture, not what we want either. Sitting says, no I don't accept your shepherdy leadership of the world, but I don't want to fight today. It doesn't matter whether Frank's owner also gets their dog to sit; in fact when you've graduated and are out for a walk in the park, you have no control whatsoever over how other dogs out there are behaving. When the situation starts, shorten up your leash, step directly in front of and facing your dog, step towards her so that she has to take a couple steps backward and get her to sit. 1/10 second sit and praise her. With practice, it will become easier to get Chloe's attention, to get her to sit, to get a longer sit, and to get some eye contact between you. By eye contact I mean she is looking at you instead of Frank. You don't direct eye-to-eye contact more than momentarily, as she would not give it if properly submitting to your authority.
When you are at home practicing, you can use whatever tools you like. The Gentle Leader (or some of the newer contraptions) and pinch collar are both tools that are useful for people who can't offer the sharp quick tug that is needed for a flat collar or choke chain. Haltis are okay too, some dogs respond fairly well to them. I would work two ways with your dog: use your crutch tools AND flat collar when you are out for a walk, flat collar in the back yard. As you progress, work the flat collar in the front yard, two houses along, eventually around the block, etc. Work at your own and your dog's pace. On a bad day (yours or your dog's), err on the side of caution.
Treats are an excellent training tool, but again a tool and you don't want to become dependent on them. They work best in association with a clicker, and unfortunately to get the best out a Gentle Leader or clicker, you need at least one session with a trainer who is familiar and comfortable with them. With treats, make sure you are only rewarding good behaviours, not using them as a lure or rewarding aggression. For instance, you may be tempted to wave a treat in front of Chloe's face to draw her attention away from Frank. Don't! What that would teach is growl and you get a reward, the exact opposite what you want to be doing!
If you have not done so already, go to your instructor and calmly and respectfully bring up your concerns. Choose a time when he isn't busy, as getting ready for your class or the following class he may be busy or preoccupied. You might need to ask him what would be a good time to talk. Focus on you and your dog, not on him or on Frank. "I am having trouble getting Chloe to listen to me. I am worried that we are falling behind in your class because she sometimes reacts badly to other dogs on leash, and we haven't been able to move past that issue yet. Would you be willing to give us a few minutes on our own, or with one other dog present, to help us make some major improvement in this area, so that we can continue to participate in this class?"
Also, you could try with Frank's owner. They have the same frustrating issue as you, and might be willing to meet with you between your lessons. Still, focus on your dog, and the fact you want to change her behaviour. Frank's owner may be under the impression that it is 100% yours and Chloe's fault, that doesn't matter. What does matter is that you would be able to trigger Chloe's bad bahaviour in a safe, controlled, and consented situation, so that you can practice, practice, practice refocusing her attention.
Don't give up on your class yet. Give yourself a chance to make it work for you.