Interesting post Lew. I believe I have a similar situation. With a number of records in my collection (possibly a majority?), if I watch for arm movement while playing a record I can note slight side to side or up and down movement. But I do not hear distortion with many of them.
Considering warps, I agree with you and optimize, there are different types of warps which may be worth identifying. The two I see most often are "dishing" where the entire outer portion of the record is raised (when viewing one side). With that, a weight or clamp may be effective, but only when the cupped side of the record is facing downward. No help at all when the cup faces upward.
The second type warp I've experienced I call the "potato chip". The outer portion of the LP has undulations. I've never tried a record ring but assume it may work if the undulations are not excessive.
Regarding hearing the warps, again I expect there are different types of experiences. Anyone who is sensitive to azimuth may notice the cupping type when tracking outer portions of the LP. The potato chip warp(s) may change tones, particularly on sustained notes with piano or voice. Of course if the warp is severe enough then mis-tracking or even the stylus leaving the groove will be heard.
Unfortunately I don't have any answers to offer from my own experience. I did obtain two 12x12 panes of glass to compress a warped record in the oven. But then I read too many failures with that to be brave enough to try.