The cooler and psu should be enough, and yes you will see some improvement.
Maybe not such a huge Max FPS bump (specially if youre overclocking your Core i5), but you will notice a decent impact on AVG FPS, and a big impact on the smoothness of the gameplay. Thats because the i7 will have more resources to deal with the request from the GPU and will be able to answer faster, so you should see better avg, and much better 1% and .1 Low (FPS).
Core i5: Base/Boost 3.4GHz/3.8GHz - 4 cores/4 threads - Cache L3: 6MB
Core i7: Base/Boost 4GHz/4.4GHz - 4 cores/8 threads - Cache L3: 8MB
But as someone pointed out already, today a 4 core / 8 threads CPU may not be a good investment, even worst if you plan on getting the newer games out there. For that a CPU with atleast 6 cores /12 threads should be the min you will want (Examples: Ryzen 5 3600 / Core i7 8700K). Unless of course, you can spend the premium of getting the real beast, the best cpu for gaming today: Core i9 9900K.
(personal opinion: avoid the 9900KS, is intel's last bad joke to consumers, I woulnt spend my money on it).