I think warm ups should definitely be a part of each class. If you leave them entirely to the student it's quite possible that new students wouldn't know what the instructor was expecting them to do, in fact. That being said, I don't see the need for warm ups to last more than ten minutes or so, and could be shorter. They should focus on getting the body limber and ready to train. They can also include some strengthening exercises such as sit ups and push ups. (Stretching for flexibility should be relegated to the end of the training session after the body is well loosened up.)
I suppose some people could view cardio training and major strength training as being part and parcel of one's Taekwon-Do training, and these things do greatly benefit your training. It's a balancing act, ultimately, with the instructor having to decide how much conditioning they want in relationship to how much time spent on actual techniques. Exercises such as push ups, sit ups, lunges, and stretching should be done by intermediate and advanced students outside of class just like actual Taekwon-Do techniques because, frankly, an hour long class 2 or 3 times a week just isn't enough training time.
Pax,
Chris