The stated purpose of "turning" is usually to introduce oxygen (which has been depleted)

The problem as I see it is that "turning" only affects some of the pile. And When you say "turn" exactly what does that mean? The term "turn" is a misnomer - the proper term is "rearrange." The only way to achieve this is to destroy the old pile and make a new one a chunk at a time. "Stirring" and "mixing" have less of a payback for the effort. Thus one feature of a proper bin is that it is easy to remove from the pile and set up adjacent to it.

Why "turn" so frequently? I am not sure fresh oxygen is that beneficial...I rearrange so as to not have a solid mass of muck at the bottom of the pile and of course the top to bottom inner to outer thing.

I have learned from this group that "turn" usually means "to mix up" or "to stir" the compost ingredients with the assistance of a garden fork. This will for sure aid in decomposition by breaking up the mass some and allowing fresh air to penetrate. However, it is a far cry from a proper rearranging, which, in an ideal world, can be described as follows, where "everyone" refers to each bit of material:

Everyone gets:

A. A breath of fresh air.
B. A new neighbor.
C. Water if necessary.

I believe the only way to begin to approach the ideal state is to set up a bin adjacent to the original (either spare bin or lift the bin off the pile and install a spare aeration chamber) and then deconstruct and reconstruct the pile a chunk at a time (using a small, agile hand fork, not a large, clumsy garden fork. The conventional advice ("top to bottom bottom to top") applies here and is simply not possible by mixing or stirring. The "inside to outside outside to inside" half can be ignored.

The word "turn", per dictionarydotcom, does not relate to composting. The closest match is definition #4: "to bring the lower layers of (sod, soil, etc.) to the surface, as in plowing." It appears frequently in Rodale and Staff's "The Complete Book Of Composting" (published in 1960 and the oldest reference I have) but the process is not described beyond the four letters.

So I am going to conclude by 1) asserting that rearranging is the best way to accelerate decomposition and that turning is much less effective, and 2) thanking those who rearrange and say so, as you are contributing to the compost renaissance.