Do Rubber Bands or Chains on Braces Actually Work?

Rubber bands have been helping to correct both Class II and Class III corrections for many dacades.  The underdevelopment of your upper jaw can best be handled with the use of dentofacial orthopedics, in order to expand your upper jaw to eliminate its underdevelopment.  This can be done non surgically even in severe cases, as you can see by looking at my cases on this site or my website.  More traditional orthodontic approaches will use surgery on the more extreme cases, but with good patient cooperation, the surgical approach is not necessary.

The short answer is Yes. Are they enough for all cases? That answer is No.

Will they work for yours? From what you describe, Yes.

You should consider that rubber bands move teeth, not bones. Moving the teeth can improve your profile balance but because they are moving the teeth that sit in the bone, not because they re actually moving the true skeletal foundation. There are only two things that can change your skeletal foundation. 1) surgery 2) lots of very light consistent pressure directly on the bone for years, if you are a nongrowing patient. This last concept allows for the same bony remodeling that occurs as we age but in a directed manner. The second is a hotly debated concept in our field.

If you would like some more upper jaw development, Google reverse pull headgear, it could be another tool in your back pocket to discuss with your ortho.
 

It sounds like the Class III problem is not overly severe and it likely could be corrected using what are termed class III elastics (rubber bands)  in conjunction with braces.  The "chains" referred to are probably elastomeric chains used for space closure in individual dental arches and would not really address the intrer arch discrepancy between upper and lower.  That would be best done by the Class III elastics.  I use the Damon system and I think we do get much better response from the class III elastics than I used to get with the more traditional braces we used before.  Every case of this type needs a very carefully done treatment plan from an experienced orthodontist who will measure and check all of the underlying skeletal structure of the jaws themselves and not just the teeth because this problem is likely related more to the underlying skeletal structure.  I do go to great lengths to avoid jaw surgery whenever possible however in some of the very severe cases nothing else witll do the job properly.

Teeth move when force is placed upon them by an outside source. That might be braces, aligners, headgear, or rubberbands. While the amount of movement is different in each patient, generally "if you wear them, your teeth will move" holds true with rubber bands for patients of all ages. The amount of movement and the results that can be produced are affected by several variables. In the situation you are describing (trying to increase the amount of overjet so you can avoid surgery), the amount of movement is affected by how big of a jaw discrepancy you have, how much the teeth have already been moved to compensate for the discrepancy, the amount and health of the bones and gums overlying the roots of the teeth, and your facial appearance. The decision to avoid surgery at all costs reminds me of the line from the first Jurassic Park movie when Jeff Goldblum first sees the island and remarks, "We spent so much time asking Could We that we forgot to ask Should We." While you might be able to move your front teeth enough with rubberbands alone, you should ask your orthodontist to explain all of the variables and why surgery might give you a better overall result.

I have to commend you for the sophistication of your question: you have really done your research!  From the information you've provided it would appear that rubber bands, along with braces, will be successful in correcting your bite.  Of course to be sure you should visit an experienced orthodontist who can fully analyze your situation and provide answers to all your questions.  In general it is a good thing to avoid surgery and your case sounds like it will not need it.  Virtually any case can be "treated" non-surgically,  but there are severe cases that are simply too extreme to get a reasonable result without surgery.  A consultation with an orthodontist will benefit you because you can establish goals and examine the best way to reach those goals.