I spent many years in live audio and have blown up more woofers than I care to remember. When ever we thought we had possibly blown a speaker our first test was always to push back and forth on the paper cone with our finger tips. If there was no movement or we felt resistance and heard a scrapping noise while pushing, the speaker was toast. If it moves freely your issue is elsewhere.
I also have a DD turntable and tonearm that are about 40 years old and have very poor damping properties and I have serious issues with rumble and speaker pumping. The first thing I did was make sure the the turntable was level. This alone improved the issue. I then purchased some isolation pads from amazon for about $8.00 for four. The ones I purchased were 4" square and are comprised of a piece of cork sandwiched between two layers of black rubber and are about 1" thick. Placing these pads under my turntable if they did not eliminate the rumble issue they greatly reduced the problem to the point that I no longer notice the issue.