I tried the bigcubes method with both 6x6x6 and 7x7x7, and I guess I can see where it could be pretty efficient. Unfortunately, I'm no longer any good at the method, so it's hard for me to tell how well it would actually work. If you had a case where all of the edges were perfectly divided on a 7x7x7 (each edge has 5 different color pair pieces on it), you wouldn't be able to make a whole edge on a single go, since you'd only be able to get 4 of them on the middle slice. But I guess then you could just do a second try to get the last one in. I did notice that a lot of times on the 7x7x7, you have 2 pieces together (usually even in the proper orientation) on one of the edges before you start, which means you usually can get all 5 at once.
I'd think that as you go up beyond 7x7x7, though, it might get really tricky to do this, since you'd almost always have to take 2 or more passes to get an edge done. You'd have to be careful not to let the confusion slow you down.
Note that with AvG, it's not double every time the cube goes up one - it's just 12 more each time. So 6x6x6 is 36 (like doing a 4x4x4, then a 5x5x5), 7x7x7 is 48 (2 5x5x5s), etc. And the nice thing is that recognition theoretically should never get any more difficult no matter how big the cube gets - all you have to do is look for the middle piece and match to it, so you can ignore all those other cubies. AvG is certainly working pretty well for me so far even on 7x7x7; I have much more trouble with the centers than with the edges.
I wonder what Erik has been using for the bigger cubes? I notice he got rather good times on 6x6x6 in the competition last week (especially considering he had a lot of pops).