This is my thoughts. I would not cut the binding off but do what I did for my DH. I found his baby quilt that was red work done by his great grandmother. When he got his divorce 7 years ago, we found it under the bed in the guest bedroom and the cats were using it as a bed. It was stained and the binding was raveled. I just took the same color material and put a new binding over the old one. That way I saved the actual quilt and gave it a new look (after I got most of the stains out). He was thrilled with the end results.