The videocard does not affect the stream quality really, unless the gameplay is at really low fps or quality settings, or having a lot of stutter.
Have actually been experimenting with this with a friend who wants to stream, he has a FX8350 paired with a RX580, and the stream quality is great, with a little tweaking for each game, since games that use a LOT of cpu are hard to stream. ( Star Citizen, for example).
You already have a 1600, this will have a lot more effect on stream quality than video card.
Both of the above cards are idiotically overpriced right now.
So get the one you can stomach the price on.
1070 is more powerful. so removing price from the question, 1070.
My buddy got a 580 for $240 USD in early November, when prices dipped temporarily and he jumped all over it.
I'd say having alerts set up with various sellers and getting either card for a price you can live with will determine what you buy.
* tip we learned. Streamlabs OBS is a LOT more efficient out of the box than standard obs in our limited experience. This may not be true if one knows all the little tweaks for standard OBS, but for us, it made a HUGE difference in cpu resources needed to feed a stream of identical quality, and had a lot more integration of features, making it a lot easier for us to use as beginning streamers.