I don't think that close quilting necessarily makes a stiff quilt with poor drape. I bought a Mountain Mist cotton batting at a garage sale. It was veeery old. The original price sticker was a couple of dollars. The instructions on the bag said to quilt in lines 1/4" apart. Close quilting is needed for a plain cotton batting, which doesn't have a scrim layer on it which is often what's used now to keep batting from falling apart. As others have said, if you don't quilt close with such a product, you will have 'migration' and a lumpy quilt. Since this type of old batting is exactly the plain cotton like what my grandma used, and the close quilting still made a soft quilt with nice drape, I'd guess her hand quilting played a part in how her quilts felt.
I know she used new fabric-- scraps from clothing she made-- so I don't think using old clothing was what made her quilts soft. However, her new fabric was not like our modern LQS fabric. If you have bought vintage fabric, you will notice it is thinner, a lighter weight, woven with a finer thread, and often fewer threads per inch making a looser weave.
On the other hand, I have a quilt I made with good quality modern LQS fabric, which I purchased LA quilting for, asking particularly for a free motion quilting design that I created, that was moderately dense in some areas to very dense in others. It has a wool batting. After the first wash, I found it was very flexible, soft, and had a great drape that I was pleased with. ...So, go figure! lol