I also used to do my binding as you described but now my hands shake so badly and I had so many quilts to finish, I wanted to get them completely done as fast as possible but still look great. Now I stitch on the back side by machine and I use only a 2.25" binding so I don't have the empty space once I turn it over to the front side. I use a foot with the flange in the center, move my needle over enough to catch the edge and stitch away. You could also use a decorative stitch. Plus, if you are using the same fabric for the binding as on the back, the stitching won't show much at all. I watched a youtube video on how to use decorative stitches from our machines. She explains which ones not to use, check for the width of the stitch, practice to see which of your feet works the best for decorative stitching, etc.
On a baby quilt, it would be cute to have a decorative stitch I would think.