Why do my bass drivers shake violently listening to vinyl

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.

Sorry noob that you had to read through all the bad advice before finally lewm helped you out. It is not the record or the turntable or arm or cartridge or anything to do with compliance or any of that. Sorry you have been so misinformed. What you are seeing is the mechanical vibration being transmitted up from the floor, rack, and shelf into the turntable. 

This happens because the signal cut into the vinyl goes through RIAA equalization that turns the lowest frequencies down 20dB. That is a lot. When playing back the phono stage has to amplify that 20dB back to sound flat. Only problem, no way of knowing what low frequency vibration is signal and what is noise. Even the tiniest vibration amplified 20dB is gonna be quite loud. Which is exactly what you are seeing. Only it is too low frequency for your speakers to reproduce audibly, but it is there and so you see it. This is the grain of truth in some of the bad advice above.

The solution lewm has in mind is to put the turntable on something more stable. There are several ways of doing this, which one will work best in your situation is hard to say. But the good news is these are tried and true simple and easy and cheap to do.

Easiest/cheapest will probably be Nobsound springs. Put the correct number of springs under each footer, will be a big improvement and might just solve your problem in one fell swoop. For $30. If not no worries they will be an improvement for sure you just might need to go further.

A nice thick butcher block shelf works well. Another one is to build a sand box. Anything the right size you can fill with an inch or two of sand. Mix the sand with just enough mineral oil to eliminate dust, pack it down, put the butcher block on top, turntable on top of that. Nobsound between butcher block and turntable. Premium isolation for cheap.

Any one or two of these will probably do the trick, and not just eliminate your woofer pumping but improve sound quality a lot in the process.