The new part here was to take all the the seasons with an all around rating of 10 or higher and see which players had the most. There have been 272 such seasons. A 10 would result if a guy had 10 runs above average in all 3 stats or any combination that resulted in 1,000 when they were multiplied by each other (the cube root of 1,000 is 10).
Here are the leaders in seasons with a rating of 10 or higher.
It seems like all these guys were or will be (or could have been except for suspected PED use), first ballot Hall of Famers except for Utley (it is also hard to tell with Frisch but it did not take long for him to get in). Maybe he should be, too. His WAR credentials are pretty good. He is 111th in career WAR among position player and had 5 straight years in the top 3 in the NL. He ranks 13th among 2Bmen right now. With 5.4 more WAR, he would crack the top 10. And his all-around ability shows he might have excelled in any era.
Now getting to Willie Mays, who is often said to be the greatest all-around player ever, he has a significant lead over Henderson and Henderson did go over 10 in 1981, a strike year of about only 108 games. Henderson had about 7.5 in the 1994 strike season and that might pro-rate to at least 10. But even so, Mays is still 2 ahead.
Here are all the guys who had 3 seasons of a 10+ rating. There are not many of those either, so what Mays did is pretty incredible.
Technical note: This time I called up all the of the following seasons using the Baseball Reference Play Index:
Seasons with 10+ batting runs, 0.1+ fielding runs, 0.1+ base running runs
Seasons with 0.1+ batting runs, 10+ fielding runs, 0.1+ base running runs
Seasons with 0.1+ batting runs, 0.1+ fielding runs, 10+ base running runs