If you have the time and determination -
1) You've decided that this one fabric is a PIDA (pain in de armpit)
2) So the decision is what to do next:
a) cover the affected areas with something else
or
b) remove and replace
3) If this is a tied quilt, removing and replacing the frayed squares might be an option.
4) If it has been quilted, I think I would go with just covering the bad spots.