Roland Of Gilead :
I don't agree with vapour that the i5 7500 will be good for gaming with a 1070ti for a few years. Those kabylake 4c/4t CPU's are already starting to get maxed out. Take BF1 for example. It will load all your cores to 100% and can cause stuttering even though you have a top GPU.
Getting the 8600k will certainly increase performance. Not only in FPS (but that will just be a small %) but also from host processing point of view. You've 6 cores to play with, and there are no games right now that will push all 6 cores to the max. But will do for a 4 core CPU.
RIght now there are two CPU's capable of pushing 144hz in most games (less so for new AAA games) at 1080p with a top GPU. An 8600k/8700k.
But, then you'd need a new mobo to go with it 🙁
By selling your current system, and putting a few quid together I'm sure vapour and I could come up with some decent builds, that would be much better than upgrading bits as you go along. Any budget in mind for a new build?
OP mentioned 60 is fine. Of course 8600k is better, but cost much more.
@ OP
I would say 7500 is still very capable:
https://www.youtube.com/watch?v=SMV2yKytQIQ
Kaby lake is not stale yet, you can postpone getting new cpus, not saying keeping it forever

New cpus are coming out every year.