A couple of thoughts; One, how about herbiciding them instead of cutting? As you're seeing, cutting by itself doesn't get rid of the entire tree - just the big portion above the ground

Leaving high stumps will not minimize re-sprouting.
If the trees are under 4" dbh, do a basal spraying, where the bottom 18 inches are sprayed with herbicide. If larger than this, "hack" with a hatchet and squirt herbicide into the hatchet cuts. Called "hack and squirt". Directions for these applications will be on the herbicide labels.
Second, sprouts from the stumps as you are now cutting will never be able to compete with the residual trees, so any re-sprouting will not affect the health of the residual stand, and will also provide browse for deer. Of course, leaving them might not be pleasing to the eye.
TSI is about impossible to be done effectively without using herbicides.