There are several things I do to help my machine bindings come out better.
(1) After sewing the binding on, I iron the binding away from the sandwich body. This makes a cleaner turn-of-the-cloth.
(2) I use Elmer's washable glue and turn the binding (and secure it) at the ironing board. The real trick is to accurately glue the binding edge to the sewn line.
(3) Only then do I take the quilt to the machine to sew the binding on.
The most forgiving stitch is the serpentine stitch (looks like a large S repeated over and over). It can be used with the left side of the S just meeting the edge of the binding, or I have also used it with the S centered over the binding edge with the stitching falling both on and off the binding. Both look good, but experiment with a sample to see which you might prefer.
Any stitch that has a straight line in it is more difficult because you need to keep that stitch perfectly lined up with the already-sewn line. If you have ironed and glued well, and if you go slowly, it can be done. I have never had it perfect -- there are always some areas that are better and worse -- but for utility use no one else notices the imperfections. Usually I use a hem stitch for this.
If you have a newer machine, you may have decorative stitch options other than the Serpentine that do not incorporate a straight line. These would be good to try out on samples. My machine is older and does not have a wide selection of decorative stitches, so I don't have any options like that.
Edit: I do think oval or circular potholders must be very difficult. Square or rectangular pot holders would be more like quilts.