M.--My husband and I used to run a martial arts school. Actually my husband ran it for almost 25 years. The way you described is similar to the way contracts used to be set up (although your story sounds like it is MUCH MORE than it should be). The problem with contracts like that is that anything can happen that would possibly make it difficult to attend all the classes within a contract period. Then you would be out of money.
The way that our school ran (and still runs, my stepson is running it now) is that you made an initial sign up payment. This would cover the uniform and registration costs. Then you paid a monthly fee. The 1st family member was a certain dollar amount and the 2nd & 3rd were half price. The 4th was free! The only additional amounts that you would pay would be test fees. You could stop and start as you needed to and the monthly tuition included as many classes as they wanted to attend. The more they attended the faster they would progress.
The key to finding the right place is to observe a class or two. Talk to the other parents. Find out how long the school has been in business and how long they have been in that particular location. There are so many fly by night places that open and close within a year's time and then open somewhere else. In the meantime they have your bank or credit card information. See how kid friendly they are. Most reputable schools will not take children under the age of 5 for a serious martial arts school. They just don't have the attention span at that age.
I think martial arts is a wonderful thing for children of all ages. They can learn so much if they are in the right school. If I can help you in any other way, please let me know. Good luck in your search.