For a given harmonic frequency, the width over time at the top few percent of it's bore axis up swing is less. There's less room for a given velocity spread to adjust load and/or tuning weight to get all bullets at their best exit angle.
That's as good as I can explain.
Ok, I'm gonna wade in here just a little bit.
You hit the nail on the head when you said that amplitude reduces with mass OVER TIME. The barrel is already deflecting downward before a shot is even fired, due to gravity, though. So, it's first movement is downward as the bullet begins it's travel down the bore. Because the barrel is pre loaded by gravity, it will come up and ultimately, the travel be larger..due to the energy held with the material. Replace travel with amplitude. INITIAL amplitude is increased, but you are right, over time it is less. We are not talking about natural frequencies of a cantilevered beam here, either and we certainly are not talking about much time.
But, at a given frequency, more muzzle deflection, initially, is equal to more initial amplitude. Forget what happens over time as we are dealing with not only natural frequencies but the forced deformation that happens while the barrel is under pressure and slightly after.
To the point...more muzzle deflection equals more amplitude and a larger arch or barrel oscillation. This is clear on target by the large out of tune groups vs. a stiffer barrel and this alone is evidence that the amplitude during the small amount of time involved is increased, not decreased. The larger radius or arch, for lack of a better word, is how mass widens the tune window. This is why a somewhat heavier tuner is best, IMHO. It both increases INITIAL amplitude and decreases frequency. In a nutshell. It slows the barrel down and makes it move in a larger "radius" at the node.
I found this interesting when doing the vibration analysis testing for my tuners because it is counter to common physics studies that mostly address amplitude "over time." But bottom line...initial muzzle deflection is increased by having a mass at the end of the barrel when fired. We measured the amount and times to the point that this was visible when testing different media in my tuner than the tungsten powder matrix that I use for dampening. Regardless of the point you are making, during the time that matters, amplitude is increased by having more muzzle deflection. The only caveat I'll add is that we were testing the affects of adding a mass to the end of the barrel. I don't see much difference though, as adding mass simply makes the barrel act less stiff.
I'm not gonna debate this because I physically measured it and one of my New Year's resolutions is to stay out of pissing matches on here. We'll have to see how that one goes..."over time."
I do like to read your posts and you are a smart man. This is very much a less than obvious result to expect if going strictly by what most research involving vibration will tell you. The key is time. There just isn't enough time before bullet exit occurs to see a net reduction in amplitude. I'm sure the numbers can vary a bit with different barrel stiffness's and the forced deformation that occurs both with and without weight added at the muzzle though. --Mike