The additions threads of the 1500X (4C/8T) would be beneficial in streaming, with the i5-7500 having the slight advantage in gaming due to it's IPC - although titles like BF1 can take advantage of the added C/T.
The difference in price between the 1500X and Ryzen5 1600 is usually only ~$20 and, although that's a lower base clock speed (3.2GHz), I honestly feel that's the minimum Ryzen chip that should be considered by most users. It also bumps you to 6C/12T, again, beneficial in streaming.
Ryzen chips really should be overclocked though, so that's a consideration. If you're comfortable with overclocking, go Ryzen.
Personally, I'd look to a Ryzen 1600 with a minor OC (maybe 3.8GHz) and a 6GB GTX 1060 as a very solid 1080p...