The i7 3970X is still a very fast, modern processor. However, it's also 4 generations old on a core basis. Since your Sandy Bridge, there has been Ivy Bridge, Haswell, Broadwell, and now Skylake, though Kaby Lake is supposed to hit by the end of the year. Technically, even my Haswell will be 3 generations old soon.
If you went from your 3970X to a 6700k, you would see single threaded performance improve somewhere on the order of 20% (minimum). In some areas and games the increase could be much more. And from a pure gaming standpoint, you also don't need those 2 extra cores/4 extra threads. Single thread is still king and very few games use more than 4 threads. On top of this, you would have newer features of the Z170 chipset and DDR4. There's an article/news post around here somewhere showing a big increase in FPS going from DDR4-2133 to DDR4-3200, some games gained up to 20 fps just from the RAM speed.
Is it necessary? No. But going to Skylake would still be a significant improvement depending on the games you play. Evaluate what games you are playing and whether or not you are happy with your fps, and look up benchmarks of those games on different generations of processors to see if the switch is worth it for you or not.
On to the 7970. I ran one for a year in 2013. They are great GPUs, but unfortunately you are a little bit wrong in your evaluation. Even one of my 380X is 15-20% faster than a 7970 (17.2% faster in 3dmark Fire Strike with both cards doing 1200mhz on air, according to the benches on my HWBOT profile.) The 7970 and 380X actually have the exact same specs but the 380X is GCN 1.2 and has faster per shader performance as well as delta color compression. I also used to have 2x 290s in Crossfire and I know without a doubt, the 290 is significantly faster. Not only that but it has double the ROPs and more texture units in addition to more shaders. The Fire Strike difference was 33.9% between my 7970 and my single 290.
On top of that, there are cards like the Fury/Fury X. The Fury X in particular has literally twice the specs of the 7970: 4096 SP, 64 ROPs, 256 TMUs. (7970 is 2048 shaders, 32 ROPs, 128 TMUs).
That's not even considering Nvidia, I will be fair and say that the 980 is around 20% faster on average than the 290/390/X, the 980ti is probably 20% faster than that, and the 1080 is at least
30% faster than the 980ti according to most reports.
So where's that leave your 7970? The first thing in the system to upgrade imo. Going with at least a Fury X or possibly a 1080 you may very well get a card that is twice as fast. This makes sense; the 7970 launched on Dec. 22nd, 2011. It's over 4 years old at this point. If it still works well for you, that's great; but if it were me, I would upgrade. $300 should get you something considerably faster.
Anyway sorry this was so long. I hope I helped, I tried to give what I thought was perspective that other posters are missing. Regards.
: I ran 1.47v through a 4770k for over 2 years because I had a dog of a chip and that's what it took to do just 4.5ghz. It still worked when I sold it and I've had no complaints from the guy who bought it.