I understand your dilemma. Sometimes the best way to save the memories of the quilt is to use it to make stuffed animals or pillows from the less damaged parts, rather than trying to restore it. If you replace fabric in an old quilt, the quilt is dated from the day the alternations were made, it's no longer "an old quilt". Does that matter to you?
I love all the ideas and suggestions given here, this is a really helpful thread. Thank you.