Why do my bass drivers shake violently listening to vinyl

As others already pointed out, the physically visible woofer pumping is typically seen when the woofer is mounted in a ported or vented cabinet that does not backload the woofer at extreme low frequencies. You’d never see it with acoustic suspension type speakers for example. ( I don’t know about Zu speaker designs or how your Tannoys are enclosed.)


incidentally we’re seated in the Kennedy Center, and my first live post covid concert is about to commence.

Just how loud is your music turned up when you experience the issue with your woofers? I also have Treo’s and have never had that problem. Just to confirm I just played some bass heavy Bob Marley hitting over 95 dB that I rarely do with no movement on the bass drivers.
The speakers are on a wood floor with spikes and BDR pads as Vandersteen recommends. My turntable is on a very heavy solid walnut wall unit and a Symposium platform and the Subsonic filter is off on my phono stage. No rumble detected.

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.