Some random thoughts. If you are a hobby-shooter, you don't need a reason to upgrade, just cash. Business/professionals have to weigh the cost/benefit analysis, and all that. Hobby-shooter? Want it? Buy it.
I was reasonably (very) happy with my 60D, until I picked up a 5Dc. I had a 28-135 hanging around from film days. It was OK on the 60D. It was an entirely different, better lens on the 5Dc. That started my migration to full-frame. (If this crappy lens is better on FF, isn't everything?) Well, mostly yes. I had a few ef-s lenses (18-55 II, 15-85, 55-250 II, 10-22), to sell off/upgrade, but the end result is much better photos, when I do my part.
Notice the collection of slow, variable aperture ef-s zoom lenses. Moving to FF prompted a foray into fast(er) primes. Which meant I could basically stop worrying about ISO. Going to f/8 (for IQ) on a 70-300 @ 6400 (on a crop) is a lot different than f/2.8 on 200L II @ 800 (on FF). Bonus? The primes (85 1.8, 135L, 200 2.8II) focus a lot faster, and more accurately than the micro-motor 70-300 and any of the 55-250s.
I don't know if EF lenses are better than EF-S. I do know that the combination of EF lenses on full frame (or APS-H) body are better than anything you'll get out of any ef-s lens on any ef-s body. Possible exception: efs 60mm vs 100mm macro. The 60 was quite useable outside the macro realm. The 100 (non-L, folks) not so much. The 85 1.8, 100 2, 135L are all faster, and focus faster. My 100 is macro only.
Upgrading your 70D to and 80D will cost you about $4-500, net. Upgrading to a 6D will cost you about $900-1000, because you should keep the 70D for pixels on target at range, and AI-Servo AF. You will miss both, if you push your 6D hard enough. Upgrading to a 5D3, will cost you about the same as a 6D, because you can sell the 70D. But you'll still be short pixels on target at range. FWIW, I like having multiple bodies/choices (currently 6D/1DIV/80D).
I am hoping my new 80D can replace the large, heavy 1DIV. I've only had the 80D a week or so, but I've already determined it can't touch the 1DIV at higher (3200 & +). Admittedly, that may be due to my lack of familiarity with processing 80D results. That said, I need the 80D for AF, not high ISO. So far, I am happy with the AF.
Good luck. Happy shooting, whatever you buy.