One thing I've been meaning to do one day is put a tenths indicator on the tip of the cutter & do a few passes at different DOC & materials. My guess is its going to deflect under load. So you set it to center reference 0.000" by whatever means, proceed to cut & the tool moves under say 0.002". No problem, pre-set it to +0.002". Now take a light finishing cut, less deflection, now 0.001" too high.
And what is the right height? When I see magnified pictures of actual cutting, it looks like quite a bit is going on at that scale. Rather complex & unpredictable interactions of shear failure, stacking, sticking, releasing, tool load, chip sticking.. and we haven't even got to heat & lubrication. My own personal data point is some of my parting tools are just plain happier at up or down a couple thou, cant remember which way. Maybe its that chip width or speed, but when I like how its cutting & check on a face cut, it does not necessarily go exactly through the center pip (or register exact to a gage datum, take your pick). So my own view is get it close with whatever device suites the setup preference, but that may not be the absolute resting pace for different permutations of materials or cutters. I guess one check in favor of a scale is it gives you that dimensional feedback right then & there. But I wouldn't buy a height gage for this. Making your own tools is usually always more fun & rewarding.