You are exactly right. I bought a Canon 24-70 2.8 L II from lens rental/lens authority a couple of months ago and it was a very sharp copy; it was tested to be 22/22 lines pair/mm. I did not have a 24-105 at that time, but, I did have a 17-40 L.
I compared both lens at 24mm and 35mm with apertures of f/4.0 and f/11.0. And like you, I used a tripod, live view, 10x magnification and wired shutter release. I thought that stopping down the new 24-70 II to f/4.0 would give it a huge advantage; not just in the corners but also in the central part of the image. Well, yes and no.
I could see no difference, at f/4.0 at either focal length, in the central say 60% of the images. Now into the borders and corners, as you noted, that is where the 24-70 II really shines.
When testing at f/11.0, the differences were not as noticeable in the borders and corners; two corners were close to a draw, slightest edge to the 24-70 II, and the other two, I would give the 24-70 II the clear/close win. Also, in the center at f/11.0, there was no difference between each lens in either scenario.
So, yes, you are correct, that the biggest difference is in the borders and corners; definitely enough so to certainly be worth/justify the extra cost to a pro/serious semi-pro for sure. The 24-70 II delivers at all focal lengths and apertures.For me, the border corner did not justify the cost/optical differences.
I would rather find and keep a good copy of a 24-105, which I now have, and buy a really good, used, fast prime in that same focal length to compliment my 24-105.
I paid $600 for the 24-105 and that leaves $1200-1300 for a prime. A more enjoyable and useful lens combo for an enthusiast like me, instead of having just one excellent 24-70 zoom.