Lord. Of course its the turntable. Specifically, its the way the record is clamped to the platter. ALL records, if you just plop them down, are going to produce some degree of the very low frequency energy that you see as woofer pumping.
When I play records, any volume, there is plenty of bass and no woofer pumping. Without filters. Horrible idea! Look at my system. Please. Took the time to post the pics for a reason: so everyone can learn. Study, observe, and learn. Please.
https://systems.audiogon.com/systems/8367Look close at the platter. Notice the black ring around the spindle? Looks like a washer. Its not. Its carbon fiber. Its hard to take a picture from an angle that shows every detail clearly so you have to really study it. Which you should do! Totally worth the time. See how thick the washer is? A record placed on this is held just very slightly higher than the surface of the platter.
If played like this, or if any record was played on this same table without the washer, just sitting on the platter, there would be all kinds of woofer pumping. Because at the scale of amplification we use even near invisible amounts of vertical motion at the record translate into a totally obvious pumping at the woofers. The fact you mention being able to see the tone arm move AT ALL tells me you’re not even close to doing it right.
So the next step after placing the record on the washer that holds it too high is to clamp it down. My record clamp is dished out inside, so as its clamped down it presses the record down onto the platter from the outside edge in. Fully clamped the record is absolutely flat on the platter. Its clamped so flat that when the clamp is released its like its been suctioned down onto the platter, which you can tell because when the clamp is released the records stay down for a second until air comes in and it pops up.
This is what you want. Its not your amp. Forget about that. Waste of time. Don’t move your table. Its embarrassing how far off track some of these ideas are. Look for a good clamping system. Clamp. Not weight. Clamp.