You are correct, damping affects the rate at which a vibration will die out in the system.. but remember, your hands and body are pretty highly damped, so while I can show you a curve of damped vs undamped vibration in an object, the reality is that with your hands on the bars, the damping coefficient of the bars themselves is pretty minor.
Having said that, in a damped system, you can have areas of effectively very high stiffness existing within the hysteresis gap between the bump and rebound curves.. and I think that this is where we get confused. So taking our damped bar tape example, if we made bar tape from memory foam (which is very highly damped) it feels great in the shop, but on the road you find that it is effectively behaving as a nearly rigid substance when exposed to high frequency vibrations. This is an example of something with low static stiffness having high dynamic stiffness when it is incapable of responding quickly enough to input vibrations. The same can be true of suspension designs as well, if you've ever experienced 'packing' on your mtb suspension fork, it is the same phenomenon, the fork cannot extend quickly enough from one hit to the next, and it is effectively 'packed' down to the bump stop as each hit is impacting an already compressed system.
Now for something like a handlebar or seat post, the amount of damping only matters if there is sufficient flexibility in the first place. This has been one of the big problems for 30+ years when it comes to carbon parts. The industry sold them on 'damping' but in many cases they were much stiffer than the more refined aluminum parts they were replacing so the damping really didn't matter, and the parts were just far more harsh riding. As our designs and manufacturing have improved, the ability of manufacturers to make products with more tuned flexibility has also improved, but I see that some of the light carbon bars of today are still on par with the light aluminum bars of 20 years ago when it comes to stiffness so the added damping might be a small benefit in a flexible enough bar, but unlikely one that would be perceptible to a rider.
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