Like you, I have the Canon 2X II and have experimented with using it on the earlier 70-200mm f/2.8 lenses (in my case it was the later IS)... I found image quality generally unacceptable for anything more than small, low resolution image uses such as Internet or 4x6 to 5x7 prints (at most).

The 70-200mm f/2.8 IS USM II and 2X III are reported to work together MUCH better and I'm sure you can find a lot of discussion about it online... probably will find some image examples, too. One of the key reasons the 70-200/2.8 "II" is superior to the two earlier f/2.8 versions is that Canon added fluorite to the newer lens (both the Canon 70-200mm f/4 versions use fluorite, too). That's one of the main factors these three lenses are described as "sharp enough to make your eyes bleed".

However, while some folks have been satisfied with the 70-200mm "II" and 2X "III" combo... some others have said it's still too much loss of IQ (and the density of a 24MP APS-C sensor like the 80D's is pretty unforgiving). You might want to rent and try the combo before committing to a purchase.

If you don't need the f/2.8 aperture (which becomes f/5.6 whenever you add the 2X), you might want to consider the EF 100-400mm f/4.5-5.6L IS USM "II". It's also quite sharp and it sounds as if you wouldn't need to use a teleconverter with it much of the time... But if you ever did, the 1.4X III would make for a 140-560mm (and your 80D is able to AF with it). The 100-400mm is probably about the same weight as the 70-200/2.8 & 2X. The 100-400 alone also would cost less than those two.

I got the 100-400mm II last year and now use it as much or more than my two 70-200s (f/4 and f/2.8 versions). I often use it with a 24-70mm and don't really miss the focal lengths in between. Mine happens to be the older f/2.8L, but if I were buying today I'd probably get the 24-70mm f/4L IS USM for it's image stabilization, near macro close focusing ability (.70X magnification), smaller size (77mm filters instead of 82mm), and considerably lower cost.

But I use both APS-C (7DII mostly) and full frame (5DII currently).... If I were only using APS-C like you, I might seriously consider either the EF-S 15-85mm f/3.5-5.6 IS USM (for it's super range of focal lenths) OR the EF-S 17-55mm f/2.8 IS USM (for it's larger aperture).... Both those have "L-like" image quality and performance while still being reasonably compact, though they have more "mid-grade" build quality and sealing.

I'm not all that impressed with the EF 24-105L "II". Image quality seems about the same as the first version, which was okay, but not great (the far cheaper, film-era 28-135mm can pretty much match it for IQ and performance). The 24-105 "II" does have much improved IS... and it uses the new "Nano USM" focus drive. But if I were shooting video, I'd wonder why Canon failed to make it compatible with the PZ-E1 Power Zoom module that they intro'd about 6 months earlier with the EF-S 18-135mm IS USM. Seemed to me a "no brainer" if upgrading the 24-105 to be more "video capable", to make it (and the 70-300mm IS USM "II") work with the Power Zoom too.

The much more expensive EF 24-70mm f/2.8 II USM certainly is a superb lens... "like a bag full of L-series primes", I've heard it described. However, it's also big, heavy and expensive. And it doesn't have IS.

You mention yours is the Sigma 150-600mm "C" and I'm pretty certain any of the above telephoto options would be a nice step up. Same would be the case if it were the original Tamron. But if it were the Tamron G2 or the Sigma "S", maybe not so much. But, of course, if you don't need and use it, why carry around such a large lens?

BTW, Tamron has recently intro'd a 100-400mm f/4.5-6.3 VC USD that might be worth consideration. It's very new, so there's limited info so far...and it doesn't come with a tripod mounting ring. But one can optionally be fitted and even with that it's half the price of the Canon 100-400 II. Sigma has a relatively new 100-400mm, too... but personally I'd never buy it because there is no tripod ring or even an option to fit one. I just feel that with telephotos this powerful, a tripod ring is an essential feature and if it's not included would at least want the option.

Thanks, I have read (I can't remember where now) that the 70-200mm f/2.8 IS USM with the 2xII combo actually performed a little worse than the older lens I have when paired with the 2xII, which is why I am thinking of jumping to the newer lens and the 2xIII. I picked up the 2XII just before my recent trip, since I knew I could not fit the 150-600 in my carry-on. The nice people at Allen's Camera outside Philly said they would work with me on the trade up after I returned. I hope to get in to see them next week. It is great to still have a solid camera store in my area.