Kurtis, thanks again for the advice -- and, in fact, for selling me one my of two Mitsubishi 7057 (Pioneer 704 clone) players several years ago. I just got that one up and running perfectly again, replacing the grip ring -- or "cushion," as Pioneer calls it on the parts site. Just a simple matter of peeling off the old ring and applying the self-adhesive one like a sticker. I will tackle the same job on the other Mitsubishi soon.
For those who may like to know, the little ring is $11.75 from Pioneer via the link in my prior post. I ordered two, and felt a bit cheated that the total cost with shipping and "handling" was over $40 for both. They came in a flat envelope. Indeed, Pioneer could have mailed them both with a single regular postage stamp. Oh, well.... at least I don't have to ship the player away and risk damage to it. That's worth the $40.
I have not yet looked into the Pioneer 503 player I have with the suspected bad load belt. That's to come.