To the OP, I'd suggest you spend time checking out the 5D Classic Admirers thread:
Ask there about who has switched from what, and specifically about the 6D (but don't limit it to that), and what their experience has been.
I do have a 5DC but don't have a 60D. My 1.6 crop body is the trusty ol' 30D, so the 5DC had some definite improvements in performance: AF is improved and definitely high ISO/low noise performance. And, the 5DC with its full-frame viewfinder is, well, so nice!
But, I also have a 1D3, and it is my "workhorse" for high performance, delivers great IQ on a par with the 5DC, and, it's weather-resistant, which can be a real plus for this outdoor shooter. My 5DC has shown little problems with just a light sprinkle, so on iffy days the 1D3 is the body of choice.
Now, the interesting thing is that I still get used to all three of these bodies. The 5DC is my walkaround body of choice, as well as my favored landscape body. The 1D3 is, like I said, my workhorse for things like wildlife/birds, sports, many events (although the 5D does handle itself quite well for a lot of things -- I've even gotten some good Birds-In-Flight shots with the 5D!
And, even the 30D gets some use! It's got better "pixels on-target" than the 5DC, and when that comes in handy and the light is good enough to get a good exposure using a low ISO, then the 30D can deliver! Lately, I've been grabbing it for macro shooting and it delivers!
As to the 60D, well, the newer cameras are certainly in a different ballpark. I don't know that I'd consider the 5DC to be "better", except that a lot of the Admirers love it for its "classic" simplicity in delivering great images. Yeah, the rear LCD is small, but you know what? My 1D3 has a nicer LCD, but I mostly use the LCD for shot reviews with the histogram. My 1D3 does have Live View, which I do occasionally use in challenging light conditions, and I'm glad I can, but still the 5DC is great for my general shooting.
I suppose if it was me I wouldn't jump that way. I'd probably hang onto the 60D until I knew that another body would really help me move forward in my photography. Sure, I'd be looking at the 5D2, and sure at the 1D4, but I don't know if I'd be looking at the 5DC except as maybe and enjoyable body to have with the 60D.
Wow, great post!