I am consistently getting significantly sharper pictures of my resident eagles and hawks in flight with my 6D over my 60D. In all cases I am using the same 400mm f/5.6 lens. I usually shoot at f/8 and 1/1000 sec or faster. I use the center focus point with both bodies, and use apurture priority with a typical ISO of 400. I continue to be very pleased with the 6D in all respects, and have no complaints about its "crippled" AF capabilities. I have never shot a 5D III or a 1DX, so I cannot compare them to my 6D. Are my results what one would expect?
"Crippling" won't affect you if you wouldn't use the expanded capabilities in the first place, and using 6d or 60d for bif indicates you have no need for difficult multi-point tracking. The 6d af will do single point center fine for smaller apertures like your f8, potential problems only occur with fast lenses when single-point won't suffice.
That's because for these f2.8 it's no double cross-point but only has non-cross precision. You might never notice if you're focusing high-contrast objects though, but on low-contrast you can get more micro misses - at least, that's my experience, and probably the reason why most other cameras have real double-cross points.
As for your 60d, it might be a afma problem - Canon removed (yes, crippled) this option from 50d-60d and re-introduced it on the 70d. If your shots are regularly off to one side (near or far), that would be it. Otherwise at least my 60d feels snappier than my 6d, ymmv.