You say new piston is badly worn with only 3 cuts. You suspect leaning out.
I would look at 3 things due to piston scoring in short period of time.
Dust getting in thru intake or the EXHAUST.
Engine over speeding or running lean.
To detect a overheating of a cylinder I happened to have a 3M infrared thermometer (costs about $50 several years ago) that I use for several things, automotive engine troubleshooting such as cylinder temp balance, (a weak or dead 4 cycle engine cylinder will be colder than a good cylinder) block temperature due to bad thermostat or temp gauge not reading correctly, bearing temperatures, immediately checking air conditioner temps, hot spots due to bad connections in electrical circuits, mainly breaker boxes to name a few.
I had a 028 Stihl that seized up when making a long time run full bar cut in a big log. (saw freed up after it cooled and the piston looked ok)
Everyone suggested the saw was running lean and I needed to use a tach and make sure it was not over speeding due to being too lean. I bought a $70 2 cycle tach checked the saw rpm's as ok and took the tach and the IR thermometer to the big downed log for testing the saw. (just happened to think of the IR thermometer for checking the 2 cycle block temp) Tach indicated no over speeding in the cut or out of the cut, but the thermometer indicated the cylinder was heating up real fast in the full bar cut as compared to other saws. The temp would start climbing real fast and I would back out of the cut so the saw would cool before reaching 400 Degrees F.
The other test saws would eventually reach around 360 degrees in a full cut and not get their fast doing the full bar cut as compared to the Stihl 028 saw that had seized.
The thermometer was easy to use by myself safely, just lay the saw in the cut and while cutting the downed log aim the red dot into the cylinder fins and monitor the temp.
A 1/4 turn in adjustment of the H jet corrected the overheat issue and the saw runs and cuts great. The saw now briefly 4 cycles for about one or 2 seconds when first placed into a full bar cut then smooths out a gets er dun.
I've checked several saws cylinder operating temp since and the average block temp of the cylinder that I have tested averages at very maximum around 340-360 degrees and temp does not get their very fast in a full bar cut with a sharp chain.
(engine loaded good and not over speeding)
You might think about acquiring a good IR temp tester and do a monitor of the cylinder temp of your saw in the long continuous milling cuts. I did not notice the saw overheating in minor cuts when the saw was not loaded up for very long. If I had not used the IR temp gauge I'm almost certain I would have ruined the saw. The carb jets were set at factory recommended setting for the saw initially and the tach indicated all ok and the saw ran great, but was too lean in the cut.