Sounds like a software (driver) issue or actually a hardware issue (GTX 970)
I don't see why i5 6500 would bottleneck, It's fine for games.
Have you tried to do a full format? (If ssd, then erase it completely using bios or manufacturer tools. That is if it's available of course)
Clean the ssd/hdd completely and do a fresh windows install, make sure to do it all with the 970 in the computer and of course don't forget to install all the latest drivers from the manufacturers official sites.

Compare the results between 970 and 760 see if it's the same deal. If it is then it must be a faulty GPU or actually something with the motherboard's bandwidth which you can try to put it one PCIE slot lower.
And if even after this you still have the same results then you need to send it somewhere to be checked. There is no other reason why would a better GPU like 970 perform worse than a 760. I would understand if a GTX 960 will perform worse, but that's because in some cases it does.
Also, going from i5 6500 to i7 6700 you will gain maybe 1 to 5 fps.. depends on the game.