Why the woofer moves badly when playing certain LPs

Use of a Subsonic filter or a preamp with one built in is almost a MUST when playing LPs. Subsonic filters were created specifically to remove that low end rumble and they work well. Just my two cents worth.

mhmeyers

Use of a Subsonic filter or a preamp with one built in is almost a MUST when playing LPs.

Such filters are necessary when there is such a profound pickup arm/phono cartridge mismatch, as in the example shown in the linked Youtube video. If the OP’s cartridge is not a mismatch for the pickup arm, then there is something amiss with the cartridge itself, such as its suspension. I think it’s preferable to treat this problem at the source - in this instance, by selecting a cartridge properly suited to the pickup arm - rather than subject the signal to a filter.

I play LPs and have no issues with woofer pumping, even on warped records, and my system is flat to below 20 hZ. But my phono cartridge is matched to the pickup arm, and I’m meticulous with setup.

Your woofer is playing the warps and surface irregularities. All that is certainly less than 10 Hz. That arm does not look all that heavy and I would think the AT would do fine in it. If you want to know for sure you have to get a Hi Fi News test record which has both vertical and lateral resonance bands and you will be able to determine exactly where your resonance frequencies are. If they are below 8 Hz you have a miss match and Ralph is correct. It could still be rumble from the lathe in which case your resonance frequency will be above 8 Hz and it would be the only possible explanation. But if it were rumble I would think you would also hear something as there are components in rumble above 18 Hz as the rumble I am talking about in certain records is more than audible and I also never turn my filter off. There is certainly a fault in those records as you say many do not do this. If you do not have a miss match the only solutions would be get a subsonic filter or don't play those records.
I use a digital brick wall filter at 18 Hz as I have an 8000 watt subwoofer system and at the volumes I frequently listen at I would probably launch woofer cones across the room. Actually what happens is the voice coils hit their end stop and make a very disconcerting bang. Without the filter you can't hear anything down there the house just shakes and you waste a ton of power until the voice coils bottom out. IMHO anyone who listens to vinyl and has a subwoofer system needs a subsonic filter. Even if they have a properly set up tonearm. Unfortunately in the analog world this is impossible to do effectively  without attenuating the bass or causing sonic degradation. But in the digital world it is no problem. What ever degradation is caused by digitizing your phone amp is more than made up for by digital bass management and room control. 

mijostyn

IMHO anyone who listens to vinyl and has a subwoofer system needs a subsonic filter.

Why?
Why not treat the problem at the source?

Cleeds, have you actually impulse tested your system and looked at the frequency response curve? Are you using digital room control? Being meticulous with setup is very important and it will certainly help but even if you have your tonearm set at say 12 Hz there is still going to be plenty of rubbish below 10 Hz that is going to get through. With the volume high enough everybody's woofers are going to flap at least a little unless they have a filter and in analog it would have to start rolling off at about 30 Hz.
It is true that sealed systems will flap less than ported ones because pressure does not build up within the enclosure. 
This may well not be a case of rumble in the record as I previously thought but there is certainly an element of record quality here and speaking from personal experience there are many modern pressings with rumble built in.