Why do turntables sound different?

I think your looking at it the wrong way. The quality of the bearing and its ability to carry weight and force quietly and effortlessly, and the drive system to move it will determine platter weight limits. Mass isn't really thought of just in the platter. Mass is usually describing the weight of the plinth required to counter the force a heavier platter spinning creates from that heavy platter. Materials used will determine how it may resonate / and how it will be dealt with. Some suspended tables actually have reasonably thick/heavy platters such as a Michell Orbe SE but many "mass" non suspended tables have very, very heavy platters that require a much heavier foundation as a platform for that spinning mass and may even require a push to start before its own weight takes over.
Cost will always have the biggest determination over choice of materials and design when someone wants to build something and bring it to market. The better materials and the combination of layering/ joining/ machining a quality , balanced platter can be quite costly. As far as the "sound" goes, perhaps you should think of how easy it is to f it all up with a mat placed on it, or how a lesser platter can be tamed with a better choice of mat. Knowing what "sound" you want usually can steer you to the type of table you should be moving toward, not the other way around.

In terms of vibration control, if you look at my turntable I have the following setup:

* The Cartridge Man Isolator (on top of the cartridge)
+ Koetsu RSP
* KAB USA tonearm dampener (silicone based arm dampener)
* HRS Analog Disk (record weight)
= record=
* Boston Audio Mat (graphite record mat)
* Symposium Super Couplers (feet for my turntable)
* Symposium Ultra platform
* Symposium Rollerblock Jr (isolator for the platform)

I am not looking at it from the right/wrong perspective. I simply wanted to initiate a serious discussion. If the mass of the platter is not determined by calculations and listening than it appears to be BS approach. Of course, it's all connected to bearing and drive. That's what I am asking - how the hell is all that bloody interconnected stuff determined? There are only a few elements but I suspect that mathematics and physics behind it should be quite advanced.
And why would Library of Congress and other establishments use mostly Simon Yorke turntables, that do not seem to be especially popular among audiophiles? I didn't hear them use Technics or Walker or that Japanese shining iron you can put four arms on.

I do not know why, but I have heard that it is like this.

Last year I spent a while in a room at the AAA-Forum in Germany
where Conrad Mas from Avid Audio made an interesting test.

There were three of his turntables equipped with the same tonearm-
cartridge combination combined with a 3 input phono stage. The
rest of the sound system was obviously identical.

So many people were in doubt  before the test if there could be more or
less significant differences in the sound of the 3 TTs.
But there were definitively interesting steps going up the ladder in his
turntable portfolio.

Can you expand on steps going up the ladder? I am pondering my final TT. Avid TTs were recommended as a possible option to consider.