The "ash" in two-stroke petroleum oil is created intentionally. It acts as a mild abrasive and is useful for removing combustion products from the engine. It is not "bad" - it is good.

Lawnboy two-strokes are massively heat-sinked. In fact, they are so massively heat-sinked that they are more akin thermally to a water-cooled two-stroke. Oddly enough, water-cooled two-strokes use "ashless" lubricant.

Hmm, see a pattern here?