Turntable rumble: what causes it?

yes it is the subsonic effect, did you use a test disc to check if the cartridge is compatible with the tonearm to achieve an optimal resonant frequency?

Rumble is bearing noise. In short the bearing does not turn perfectly smoothly. You don’t mention hearing anything so might not be that. What you describe is woofer pumping. Usually caused by environmental vibration- wind, machines, people moving around, etc. The information in a record groove uses RIAA equalization so that the lowest frequencies are down 20dB. So whatever vibration your platter is doing the phono stage amplifies by 20dB, and that right there is a lot. But then depending on your cartridge the equalized signal is then amplified another 40 to 65 dB, just to get it up to line level. So it does not take very much vibration to make the woofers pump back and forth like that.

So, which kind of vibration are we talking about? Bearing rumble comes from the bearing. The further away from the bearing the more the platter physically amplifies this. So if your woofer pumping is really bad at the beginning but almost gone by the end of a side then probably this is bearing rumble. But if it is pretty much the same all across a side then probably it is woofer pumping from environmental vibrations.

Yours being vintage the bearings can go dry and rumble, and the suspensions weren’t that good either (and often use materials that get hard or even fall apart with age), so odds are about 50/50 with no reason it has to be either/or it could easily be both.

Take the thing apart, clean (if possible) and lube. Then step 2, put the table on Townshend Pods. Or the budget solution, Nobsound springs. Lube and springs will not only eliminate your woofer pumping they (especially the Pods) will also have your table sounding better than ever.

how far are your speakers from turntable?
for vintage turntables rumble filter is highly recommended so you might consider either add-on or phonostage with rumble filter.

The turntable sits on an antique wrought iron table with glass top.  It sits about 5 feet from the speakers.  

BTW the table does have a decent suspension, and weighs about 25 pounds.

There are long threads on "woofer pumping", and there are several possible causes.  You might do a search here and on Vinyl Asylum (go to Audio Asylum and select Vinyl Asylum from the list). VA has a good search engine.

"The turntable sits on an antique wrought iron table with glass top. It sits about 5 feet from the speakers. "
I saw this post from you after having submitted my last post.  At least by description, your set-up sounds like a set-up for woofer pumping.  Not only is it possible that your tt is too close to the speakers, but glass is not an optimal shelf material for a turntable, suspension or no suspension.  Glass is highly susceptible to environmental influences especially including airborne energy from the speakers.

The turntable sits on an antique wrought iron table with glass top.

Yikes! Won't hurt to lube the bearing. But ...  wrought iron... glass... there's your problem. 

it is possible that even the feet of the turntable, if they also have rubber, this has disintegrated and no longer acts as an isolator.

However, I believe that the Pioneer is completely underpowered compared to the rest of the system, even if the convenience of the automatisms it possesses can be indispensable.