I go by remarks from Barbara Brackman who is well known for her research on antique quilts and her reproduction fabric lines. Barbara has stated that when sewing machines were selling at over 100,000 per year by 1861 it was not uncommon for the backing of a quilt to be pieced by machine even if all other seams were sewn by hand. To me, it would seem to follow that women would also sew the bindings on by machine, since the seam would not show. Quilts prior to 1900 were also machine quilted -- with curves, by the way!
One of my favorite expressions is,"Women were not stupid, they were put upon!" (meaning they often had to do a phenomenal amount of physical labor to care for their homes and families; of *course* they would avail themselves of any labor-saving device or appliance they could!)
Sew the binding on by machine and feel no shame about it. Go for it in good company!
Jan in VA