$100 says none of these believers will ever post a measurement or other proof showing the affects of vibration.
I've posted this before Ethan, but this is a long thread and I don't blame you for not wanting to wade through it.
I own a Scully LP mastering lathe equipped with a Westerex 3D cutter. The lathe itself has 3 feet which are carefully machined stainless adjustable points. They rest in turn on an anti-vibration platform in machined cradles; the platform in turn employs softer feet and rests on top of a table built for the lathe and platform. The table also employs pointed adjustable feet just as the lathe itself does.
Clearly this is an attempt on the part of Scully to control vibration issues and for obvious reasons. It has to be able to make a silent cut on the lacquers it cuts. You might be interested to know that its quite good at that- a lacquer cut on the lathe is so quiet that no matter what your playback electronics, they are the noise floor and not the groove itself. That puts the groove noise easily at about -95db. Obviously things are different when the actual LP is produced! Apparently the pro audio industry recognized the possibility that vibration could be a problem and dealt with it in many of the same ways that audiophiles are using today. My lathe was made about 1950 or a little earlier, so these techniques you might say are 'time honored'.
(as a side note, Acoustic Sounds (Salinas, KS) recognized that vibration was playing a role in the surface noise of the LPs they were producing in their QRP (Quality Record Pressings) facility and so modified their pressing machines to not vibrate whilst the LP was cooling within during the pressing process. We've done a couple of projects through them and the noise floor is impressive- very nearly that of the lathe cuts themselves.)
A further example might be an Ampex 351 tape electronics chassis, which incorporates isomeric isolation on each of its circuit boards. Clearly Ampex engineers were concerned about the effects of vibration as well.
The Marantz 7C preamp employs an isomeric isolation technique on the subchassis of the preamp. This is clearly done to reduce vibration sensitivity.
We use isomerics to isolate our circuit boards in two of our preamps as well as intentionally dissimilar resonances in the various parts of the chassis which cause the chassis to be quite dead but I am assuming they don't count in your challenge.
A further example of vibration awareness in the industry is demonstrated by Technics in their SL1200G turntable which is not a reissue of their older SL1200 although it looks like it at first blush. But in closer examination, it is seen that the turntable employs 4 different methods of reducing the effects of vibration including damping the platter itself.
These five example easily show that vibration has an effect and are the proof you are asking for. I nor anyone on this thread is expecting that you will actually come through with the $100 but if you are your word then I am easily enough contacted. Of course, an out for you is that I'm not a 'believer' so much as a 'knower' and there is a profound difference! Your quote does refer to believers so I am probably disqualified.
Incidentally it appears that your comment about LPs is misleading; the distortion LPs convey is much lower than you seem to realize. For example, the cutter amplifiers used to drive a cutter head usually are spec'ed to make about 10x the power needed to completely fry the cutter head itself (obviously this means that the mastering engineer has to be quite careful as the cutter head is thus very easily toasted). In essence, the cutter amps are/were loafing in order to make any cut ever put on a stereo LP. The problem with LPs is not built-in to the LP itself as it is the often poor setup that occurs in playback (which is not the fault of the LP) and also the much higher distortion that occurs if analog tape is in the equation. So I am making a distinction between the capabilities of LP as opposed to the tapes that are often used in making them.