OPTION IF YOU GOT MONEY:
I would say if you go gaming and watch YouTube/Netflix or anything like that on a second monitor I would go I7 7700K. If you are going to go and play games and watch YouTube/Netflix and plan on doing streaming or recording for a service like YouTube, I would then say go Ryzen 5 1600X.
COMPARING THE TWO CPUs:
The Ryzen will help you in things like streaming due to the fact that it has 6/12 (6 cores 12 threads). The Ryzen is also not a slouch it will perform close to the I7 7700K in gaming, may be 5-8 frames less in some games and same in others with some the Ryzen pulling ahead. I would also say the Ryzen will have a better upgrade path, so if you go Ryzen you will be able to upgrade in the future, while if you went I7 7700K you will then need to upgrade MOBO and CPU again.
Now the I7 7700K is better in things like gaming, it wont do anything watching videos really, may loose 2 or 3 framed, but thats nothing. The I7 7700K is also not that bad at streaming to be completely honest. I have a streamer buddy and he streams on his I7 7700K and GTX 750 TI and he has zero problems streaming in 720P 60 frames (he gets 5-10 FPS drops not much), he also streams in 30 FPS and really gets no noticeable drops, may be around 5 at absolute max, unless the game is very CPU bound it will then be more.
CONCLUSION:
So in conclusion with what you stated I would go Ryzen, not because it performs better because it does not, but because you will have upgrade path in the future (due to the new socket). You can always go I7 7700K route and upgrade MOBO and CPU in the future if you got the money for it, but if you dont got a lot of money I would stick to the Ryzen upgrade (Ryzen 5 1600X).