Im with PC Tailor, but I can talk from experience. I use to have a Core i5 3570, yeah I know is a few generations old, but it has the same base and turbo frecuency (and rest of spec too like L3 cache). Used to also own a GTX 1060 6GB DDR5. They used to work good togheter until I started to playing newer more CPU demanding games like Shadow of The Tomb Raider and Battlefield 5.
On SOTR (1080p@60Hz, vsync ON to avoid flickering, with every detail on highest but for hair and shadows) everything was sorta smooth, but the game stutter very often. Using MSI Afterburner you could see it was CPU bound with over 90% ussage all the time. (avg FPS around 50). For this kind of game wasn't ideal, but it was playable.
On BF5 (1080p@60Hzs and almost every detail on high, some on ultra, but for shadows) you also felt the stutteting but this time it last longer, it was more annoying when it happend, avg fps werent that bad unless you were in crowded areas.
After a while I got an RTX 2070 and the things really didn't improved too much.
On SOTR (same config than before) nothing really changed, yeah I was getting a few more AVG FPS, but the game feel the same crap than before (I knew this was going to happend anyways). Now my Core i5 was 100% bound all the time.
On BF5 (same config as before) everythig was runing like with the GTX 1060, a little bit more avg FPS, but 1% low and .1% low were still very "low". and also the CPU now was bound to 100% too.
The RTX 2070 on those case was never over 50%, maybe if I had a high resolution display or high refresh rate panel it could be a diferent story (and yes I also know the RTX 2070 is over-overkill for 1080p@60Hz, but Im working on gettting a 1440p display soon).
In short as PC Tailor wrote, it will depend on the game, but keep in mind that in this two games my Core i5 was even bottlenecking the old GTX 1060, so nothing really improved with the RTX 2070.
After 2 months I finally was able to change my CPU and man, that was a huge impact, no more stuttering, cut scenes never froozed again, everythig on the scene was loaded at the same time and enemies didn't spawn at your side without you noticing it lol. Everything feel smooth now.
Cheers