Do Rubber Bands or Chains on Braces Actually Work?

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.