The 24-70mm is a great lens, but it often whiffs on focusing. I'm amazed at how many more keepers I have with my 70-200mm and my 17-40mm.
Well, the 17-40 is a f4 lens and uwa at that, so it's not that hard to get something into focus :-p ... as others wrote, maybe you should get the lens serviced because the L1 is known for being more fragile than the L2 if it ever took a bump.
Am I fooling myself that my current 24-70mm will perform better on the 6D?
Yes, you are fooling yourself at least partly: One of the most important advances of the 24-70L2 is the more precise af motor, but *only* the 1dx/5d3 (and now probably 7d2?) can make use of it.
So basically you're paying for tech you cannot use, that's why the Tamron 24-70L might be a better combination - sharper than Canon's L1 and with IS, plus having a fast lens af doesn't really matter with the 6d anyway
Due to finances, I would have to sell the body or lens to fund the new purchase.
Imho stretching your budget to get top notch glass on a crop is complete overkill, even though investment in lenses usually is the smarter choice. The f2.8 standard zooms are really designed for full frame for low light and thin depth of field - if that doesn't matter you could simply go for the f4 zoom otherwise.
You didn't write what you usually shoot, but with your L1 lens and the 6d you'll see a vast difference, while with the L2 it'll take some pixel peeping or very large prints to notice the difference.