Why do my bass drivers shake violently listening to vinyl

You already have anti vibration feet designed by Technics for their new G series. The cabinet is a 4 layered construction. You DON’T NEED any springs or butcher blocks or any platforms under your turntable (these crap is for belt drives).

Technics turntable can be used by PRO near a HUGE sound system with tremendous bass response, probably 1000 times higher SPL than from your speakers.

G and GAE are top model in SL1200 series and stock feet are very well engineered, but if you want to try something else you have to check Isonoe (they are made for Technics).

Read this topic.

Unless you are not placing your turntable right on a SUB I think the problem is not the feet.

Chak, If the 1200GR is without construction or operational flaws, I quite agree with you.  If it has a construction defect, like an off-balance platter or bad bearings or something rubbing on something, then it is better to identify that problem than to put a rumble filter bandaid over it, I think. For that matter, I am guilty of suggesting other types of bandaid, so my own advice is also flawed in that way.  One would like to examine the TT in isolation.  The only thing I suggested that might help to eliminate problems in the Technics is to temporarily replace it with another known good TT, and determine whether the woofers still act up.  If that test is passed, then maybe the rumble filter is the way to go or any of various forms of enhanced TT isolation first.

Vinyl has much less bass extension the digital, limitation of the medium.
Vinyl playback equipment often has more. Not from the program. There used to be subsonic filters to cheat your way out of it but now... 
Saddle up.

That’s why there’s been some mention of the KAB rumble filter which is a well made add-on.

A rumble filter will definitely eliminate woofer flutter. But it will do nothing to improve sound quality. Springs, sand box, etc will definitely improve sound quality, and also in most cases eliminate woofer pumping. You can spend your time and money on improvements, or fixes. 

The low frequencies being put out by the speaker are traveling along the floor, into the rack, picked up and amplified by the cartridge and phono pre and sent back out the speakers further exacerbating the issue?

No. That is different. There are two main forms of vibration to deal with, both vibrations but completely different in effect. Your original question is about subsonic vibration. Speakers can't audibly output, you can't hear it, you're concerned with something you can see but not hear. 

Then there are vibrations from speakers in the audible frequency range you can hear. This range of vibration will smear detail and color instrumental timbre making individual instruments sound less distinctive. Putting speakers on springs like Nobsound will greatly improve this. As will putting the turntable on springs. They will not eliminate the large amplitude low frequency pumping you are seeing. They will however greatly improve clarity, detail, dynamics, imaging, etc.

This is why I recommend the things I do. Springs, mass, etc work together to solve both problems. Solve. Eliminate. Not patch over with a band-aid. While simultaneously improving sound quality. None of which you get with a filter.