Do what you thought. The new backing fabric will strengthen and support the older top. You can cut a small piece of the old backing and stitch it on the new backing with a label to say that is what the original backing looked like.
I am all in favor of "renewing" old quilts so they are solid and usable. All too often if you don't, someone will come along and say "what is this old ragged thing?" and toss it or use it for changing tires. Too sad. (I rescued one being used as a tire changing pad.)