Bringing the backing to the front is the way my grandmother always did it. I have an old baby quilt that my great-grandmother made and it has a "knife-edge" binding where they tucked both the back and front in and then top-stitched it all the way around. I have done traditional binding, back to front wrap, and pillowcase or birthing style.
On the first quilt I ever completed, I didn't know how to make binding and didn't have any quilts around to look at so I measured my sides and made "cap-stone" like things to bind the edges--like a waistband on skirts or pants. It worked out somehow but was definitely not the easiest or best way to do it! Why I wasn't smart enough to bring the backing to the front, I have no idea.