I'm not saying that balancing should never be accomplished. But if pilots are experiencing MAJOR vibrations from what they suspect to be an out of balance tire, I would be questioning other things that you may be covering up by balancing.
Everyone like stories so here's my inverse story to yours. Our bonanza that we purchased and flew for a while before deciding to change out the tires had no noticeable vibration, never questioned it. The tires had been previously balanced, and since the paint was in bad shape, I decided to completely strip them down to bare metal, inspect, repaint, new tubes and tires (of high quality) and ended up changing out most of the bearings and a couple of the races. I pulled the balancing weights off because I needed to strip it down. I reassembled the tires properly. Re-installed WITHOUT balancing and have been flying for 2 years now with no vibrations.
Now my argument is all the other factors besides the tire being out of balance are:
-Quality tire and tube of the correct size
-Condition of bearings and races (Corrosion on the races, flat spots on the bearing, slop in the bearing, etc)
-Condition of the wheel itself-Installing the tire and tube correctly (did you line up the valve stem in the right spot? Are there any folds in the tube?)
-Installing the wheel assembly onto the aircraft properly. (proper torques, does the bearing fit properly on the axle itself?)
-Is your shimmy damper in good condition? Condition of your struts, attachment points, hinge joints, play in the landing gear,etc (slight vibration causing larger vibrations?)
If any one of those above things are incorrect and causing your tire to be out of balance, isn't balancing just masking? If everything comes down to being correct, and all the above looks good and is right, then there may be an imbalance of your wheel itself. The quality of your wheel could be poor or the indexing is off. I would check the balance of the wheel by itself. Maybe by rotating the wheel halves over a hole or two (as long as it's not an indexed wheel) it then puts your wheel itself into balance. Don't you think the manufacturers of the wheels and tires are concerned about balance and provide products that are at least damn close to being balanced? The aircraft I work on for work is extremely sensitive to vibrations, takes off and lands at 100mph and we do no balancing of the tires and have never noticed or have been concerned.I think people obsess over the idea of balancing, and may be over looking other issues that by balancing you are masking. Now with all those things done, and your wheel is just out of balance and there is nothing you can do besides buying a new wheel, then by all means balance (motorcycle shop would probably work out the best.) But I don't think it's a quick and easy YES BALANCE answer. If one of my clients was concerned about vibrations, there would be a lot of other places I would look first.
Justin
A&P/IA