Assuming you have a decent amount of money to drop on a GPU which if you are considering the 1070 you probably have plenty to spend I would consider some of the new GPU's that are coming from AMD or one of the GPUs that is already out the R9 Fury.
The GTX 1060 is not extremely future proof while the 1070 is definitely future proof. Airing on the side of price though going with something from AMD (If you're willing to take the leap) will be cheaper and provide you with very acceptable performance on most games. As far as shadowplay goes there are plenty of programs that can easily replace it and make it not that much of an issue. With you wanting to do 1080p most mid-range GPUs even a generation back are fine with what you want to do, but bear in mind that DirectX12 is becoming more relevant in newer games, with that being said AMD GPUs are very good at that while both the 1060 and the 1070 are not great at it.
Overall you would do well with something like an R9 Fury or a GTX 1070 for gaming at 1080p 100+ fps on most current games and most new games of 2017 and onward you would be in the 80 fps range. For you, I'm guessing that price and user friendliness are also important and the Fury would cost you almost half of what the 1070 costs (It goes on sale for 260 or lower all the time) and the 1070 would offer you 8GB of VRAM. So it all comes down to what you personally enjoy using, but bear in mind that in the future if you plan on adding a second GPU you'll want a monitor with adaptive sync and getting G-sync will cost you an arm and a leg, and probably a kidney.