I agree. For playing games a i3 will do a better Job since its ipc is higher (interactions per clock). AMD i would prefer if there is a need for their multitask capacities.
My Suggestion for a new build decent gaming pc:
i3 6100 + ASRock B150 Board + 8 GB DDR4-2133 + a GPU according to your requirements (RX 470/RX 480/ GTX 1060) + a 400-450W PSU providing more than enough power to handle everything + ssd/hdd according to your wishes.
In most cases the box cooler will do a pretty good Job of cooling the cpu but if you prefer some custom solution i would choise a 15-25€ (dont know what is the equivalent in your currency) Tower CPU Cooler, like the ekl ben nevis, be quiet pure rock, thermalright true Spirit or any other in that category.
I would never build again a pc without ssd but if the Budget is low you maybe could simply use a hdd instead.
Did i miss anything? feel free to ask me via pm for more help
greets J.