Why do my bass drivers shake violently listening to vinyl

This is what sub-sonic filters are for.

sounds to me like there may be a very low frequency feedback loop.  you may not hear it at the distance or volume you are listening.  the structure may be responding and resonating 20 - 30 feet away.  your mechanical pick up on the cartridge, tonearm and turntable may be picking it up and there's the loop.  Streaming has no mechanical pick up so it would not be as likely to create the loop.  if you see long, consistent, excursions of your woofer there's a clue.  It's probably not very good for the surrounds of you woofer especially if it's long excursions.  people don't hear 20hz very well.  it has to be in the 70dB range for most people to even pick it up.  you can almost see very low frequencies better on your woofer since the cycles are slow enough.

Is this some kind of feedback loop causing this?

Yes. Use a sturdier furniture piece, add slab of granite under the turntable. Consider upgrading the feet on your TT to something with more effective isolation. Your problem will disappear.

We have seen this problem hundreds of times over the decades. The problem is NOT in your speakers. We believe as many others here it is compliance problem. The simple solution is to use a rumble filter, if you really want to find out what the problem is here is what we suggest. Take just your turntable to a completely different system and play it with several different albums, if it still has a rumble problem then the problem is in your turntable, if not it could be your phono pre or interconnects. If the problem exists in your turntable, first check your tonearm for problems like loose bearing or places where it maybe hanging up along it's arch also check your setup parameters on your cartridge to make sure they are correct ( We always suggest using a test LP on any setup) If the problem persists change the cartridge to one that is more compliant to your tonearm (check your setup again) If the problem still exists then it would have to be in your tonearm. Many times we had to change the tonearm to eliminate the problem, as strange as it may seem there are times when we changed the tonearm with the exact same model tonearm and it solved the problem, we refer to those pieces of equipment as having "spooks" in them, we ship all those pieces to the X Files. Good turntable isolation cannot be understated and definitely improves the quality of vinyl playback but we have placed turntables on solid concrete platforms and they still had rumble problems, how can that be? Don't know but it happened. More X files stuff.

Two questions for arrowhead: (1) By what mechanism do you think a phono pre or a pair of ICs can cause rumble (except in the case where the phono stage has very low bass frequency extension, enabling it to transmit rumble frequencies to the downstream components, but that's where the rumble or subsonic filter works)? And (2) why do you focus on the tonearm and not mention the platter bearing as a source of rumble?  I do agree it's a nice idea to take the TT to another system and determine whether rumble follows along with it.  I had suggested inserting a known good TT into the OP's system, for the same reason.