owning both a 7D and a 6D and having spend a lot of time chasing birds and birds in flight, I would quite confidently say that its the operator of the camera that will be the limit, not the camera.
You will not get suddenly get better action photos with the 5DIII than you do with the 60D unless you learn its strength and weakness.
On your 60D, if you use the Center focus point, not spot focus, put it in AI servo take your 70-200 and go in a park. Try taking photos of little and big birds, in particular birds in flight.
Would be awesome if you could post some......
Its more about technique and practice and having a fast shutter speed and giving the AI Servo a contrasting surface to work off and keeping the bird in the center of the photo.
Or focusing on the eye with one of you other focus points to frame the bird nicely. Even if it isn`t a cross sensor, picking a different focus point and putting it on the right spot typically the eye, makes all the difference.
When you get good at it with any camera body and you have taken a thousand photos or so, you can then appreciate the subtleties of the various system.
What I can tell you is that the 6D cross center point, is more confident and does a slightly better job of tracking a subject than the one in the 7D, but I tend to use the 7D for bird photos.
Mainly because of the crop factor I get more bird into the photo and I can pick which one of the 19 cross focus point I want to use. Unless the light is a bit poor and then I take the 6D.
Once you chase birds anything else that moves becomes easy, because its much bigger and slower. So basically I wouldn`t buy a 5DIII purely due to the fact that is has a " better " focus system.
Generally speaking you will not suddenly get photos on a 5D III that you missed on the 60D unless you have got the skills, you will not get the full benefit unless you experiment with settings and fine tune the focus system to suit and understand where its strength and weaknesses are.
If haven`t already done so maybe read up Gordon`s reviews on the 5D III, 6D and 7D in particular the bit on the focus system.
Canon Powershot S95, Canon 6D,7D, Canon 40 2.8 STM, Tamron 24-70 2.8 VC, Canon 17-40 L, Canon 15-85, Canon 85 1.8, Sigma 30 1.4, 50mm 1.8, Canon 100 2.8L Macro, Canon 70-300L +Kenko 1.4 Pro 300DGX, Canon 430EX II and RS 4 Classic