Why would he need new RAM? I upgraded from an i7-6700k to a Ryzen 7 1800X, and I changed only CPU and mb. It's true that Ryzen favors fast RAM, but it's not a must. I think both options have merit, getting a 7700k or a Ryzen 7. I would discard a Ryzen 5, with a good board it would be more expensive than the i7, with less single thread performance and only 50% more threads. Additionally, if you get a Ryen 7 1700, be prepared to overclock if you want gaming performance comparable to the i7.
Anyway, I agree with 4745454b and WildCard999
4745454b :
Streaming while gaming will be better on the Ryzen platform for sure. Even updating your bios and upgrading to the 7700K will give you issues here and there. The 1600 or Intel 8xxx will be better for sure. I'm not sure it's totally worth it considering you need new everything, including ram. If you are streaming though you don't really have a choice I guess.