From my testing over the last 10 years with amateurs in my schools and professionals on the PGA Tour the saw grip consistently provides superior results for putts inside 10 feet and good results for putts from 12 to 20 feet. But performance on long putts tends to be worse than with conventional grips.
Granted this assumes that all putts are hit with perfect speed which for most purposes is one that carries the ball 17” past the hole in the event of a miss. Hit any harder the ball on the “high” line might lip-out. And with less speed the low ball would be more susceptible to the “lumpy donut” of footprints around the cup and tend to miss low.
Although the saw grip falls into the “alternative” or “unconventional” category it may be the most natural hold of all because with the saw grip your right hand never strays from the correct impact position from start to finish.