I'm enumerating corners-only, tredge-only, and centers-only disturbed positions. The one with the fewest instances per depth is the worst to solve last. As it turns out, this will be the corners.
Theoretically, solving corners first, tredges next, and centers last should produce, on average, the lowest move counts per solve.
I'm not sure if this way of looking at things is right or not, but I do know it doesn't take all factors into account. When talking about what order to solve groups of pieces in, you also want to consider what kinds of moves and move sequences are allowed, and what are not, based on what you have already solved and must keep in place. The question to consider for each piece type is, how many moves do I lose due to having to respect the earlier steps?
Of course, usually this number can't be known for sure, but it is something that can be reasoned about. For instance, although there are 4-move centers only cases, they only affect six centers out of 48 (and in a very specific pattern, too), whereas you could otherwise do something like r U b and affect many centers at once in a few moves - giving a pattern that would require many more moves to solve if you know you can't disturb the edges or corners. So one could argue that the restriction of having all the edges and corners in place could potentially cost a lot of moves. With corners, on the other hand, you are looking at 11 moves to solve them first and maybe 18-20 to solve them last. Are those 7-9 moves worth that restriction in movement for the whole rest of the solve?
However - corners last should really never be considered. That's because they should always be solved while solving the middle edges. I'll explain.
There are really two fundamentally different types of edges, and lumping them together as a "tredge" can obscure some important points. I'll call them middles (the one in the center of each edge, which only shows up on odd-layered cubes) and wings (the rest). The two big distinctions are (a) middles have two possible orientations (and thus can be flipped) whereas wings only have one; and (b) like corners,
middles can only be moved using turns of the outer layer (relative to the fixed centers). The net effect is that middles can be treated like 3x3x3 edges.
This matters because, as any 3x3x3 solver knows, solving corners first and then edges, or edges first and then corners, is slower and takes more moves than just solving the whole cube directly. So if you think of the tredges as two separate steps - solve the middle, and, pair the tredges (i.e. get each middle next to its two corresponding wings) - then it immediately makes sense to combine the "solve the middle" step with solving the corners, in a 3x3x3 stage, like we do in reduction methods. Then really the three steps are solving the centers, pairing the tredges, and the 3x3x3 stage, and now it makes more sense to argue about how to order them. And of course, the 3x3x3 stage can be done in the same number of moves at any point in the solve, so you might as well do it last.
