It scores for progress toward known folds. Known folds have enough stability to fold. I think right now there are only just over 400 known folds. I don’t see how this wouldn’t be expected to correlate with stability even from just a sequence perspective.
Obviously, but your problem is not broken proteins that kill an organism, it’s the build up of many many almost broken proteins that still function well enough to keep an organism alive and well enough to reproduce. I didn’t post those links just to show that snps often break proteins. I think it’s a pretty safe bet that somewhere between being completely denatured and the original stability, there is an area of less stability that is still viable. But you knew this or you wouldn’t have brought up neutral evolution.