I have stuck my oar in on this plenty of times.
I have cured numerous cases of nosewheel shimmy by simply dynamically balancing that nosewheel, without touching anything else. Dynamically, not statically, which is about all any airplane wheel gets. If it isn't dynamically balanced, all those expensive parts can only mask the problem for awhile until they're worn out again in short order.
Finding someone to dynamically balance it is the problem. Motorcycle shops often have a dynamic balancer that will fit a nosewheel. The seals and bearings must be out of the wheel. I built my own balancing machine when I went to work in another shop, based on an old machine that I had converted to do nosewheels in the flight school.
You have to think about it. When you buy new tires for your car, what do they do to them before they install the wheels back on the car?
They dynamically balance them. That wheel has to be spun to determine the amount and location of the imbalance, and simple static balancers cannot do it, despite what some vendors of static balancers tell you. I, and a lot of other old guys here, remember the days when tire shops had only static balancers, and the steering wheel would shake constantly in cruise, with some speeds being very bad. ANd that was in cars, where the suspension and steering stuff is far heavier and rigid than that in an airplane. The shimmy damper in the airplane is just an attempt to stop shimmy caused by dynamic imbalance and dates from the old days when static balancing was standard practice. Cars NEVER used any shimmy dampers at any time, and dynamic balancing eliminates the need for them. It's funny how we complain about the archaic technology still being used in GA aircraft, and then go and use archaic methods to try to cure shimmy.
Do a search on POA for nosewheel shimmy. I'm not going to redefine it here anymore.
If there are guys posting here giving advice on what to fix, and they have never done a dynamic balance and experienced the result, their advice is badly lacking.
See this:
http://www.aircraftmaintenancespecialties.com/tech.php?id=3
TLDR? An excerpt:
Nose Wheel Tire & Wheel Assembly Balance
Balancing the nose wheel/tire assembly is the most important point to check in trying to solve a shimmy problem. Aircraft tire and tube manufacturers paint a red dot on the tire for the "light" spot. A yellow stripe on the tube, or the valve stem should be aligned with the red dot for coarse balance during tire buildup. Then the tire/wheel assembly is balanced with a static type (bubble) balancer and generally does an acceptable job.
However, the preferred method, and sometimes the only method that can solve persistent shimmy problems, is dynamic balancing. (Dynamic balancing is when the wheel/tire/bearing assembly is spun and the proper weights and locations determined electronically). Very few light aircraft maintenance shops have the capability to do spin balancing, so AMS suggests that local motorcycle shops be contacted until one with a spin balancer is found. Usually these shops have the necessary mandrels to mount an aircraft tire/wheel assembly. Balance is achieved by affixing lead weights to the wheels.
Bold emphasis mine.