I tried that with freezer paper one time - silly me! Of course there was no way to get it out of the seams. It was just a little experiment with a few 1 1/2" squares. It did make a very tidy grid. I could make a book cover out of it maybe.
More recently I heard of a fusible version of Stitch N Wash, which is used in machine embroidery. I have not tried anything like that, but it seems to have potential for this, doesn't it? Not all of the fibers in it wash out, but maybe enough to make it softer in a quilt? It's just an idea that I might try some year. Thought I'd mention it in case anyone happens to have some of that on hand and an impulse to test it for us.