theres a 6 core 6 thread i5 yeah but it complicates things more, id personally build ryzen but its down to you
the cooler you picked is great but a hyper 212x will allow you to overclock the same, ryzen is quite good with overclocking
plus ryzen really benefits from faster ram, ive thrown 2666 on there, and it works out cheaper than your budget
https://pcpartpicker.com/list/QFNM9W