AMD or INTEL
Does $300 include the motherboard? I'm also assuming you mean $300 USD.
If $300 does not include the motherboard, $300 could get you an Intel Core i7-8700 (6C/12T, locked) or an Intel Core i5-9600K (6C/6T, unlocked). These are both excellent CPUs for gaming and should easily allow you to make the most out of that GTX 1070.
Alternatively, you can get a Ryzen 7 2700 or a Ryzen 7 2700X. They're both unlocked, so the difference is their base clock speed and included cooler. The Ryzen 7 2700X, out of the box, will be better than a Ryzen 7 2700, and it will come with a beefier. However, if you buy the cheaper Ryzen 7 2700, you may be able to overclock the difference and use your savings for a better AM4 cooler.
That said, gaming performance of a Ryzen 7 2700X isn't going to be very much different from the Ryzen 5 2600X. Value-wise, the Ryzen 5 2600X is much better for gaming.
If $300 includes the motherboard, then I'd get a Ryzen 5 2600 or Ryzen 5 2600X. The 2600X is slightly more expensive, but it comes with a better cooler and a better out of the box clock speed, similar to the the difference between the Ryzen 7 2700(X) CPUs. As such, even if you get a 2600, you may also be able to overclock the difference, too. B450 boards are a bit cheaper than Z370/Z390 boards, and you should be able to find a full ATX one for under $100.
You can get Intel here, too, but if you're looking for a recent unlocked CPU, you're looking at the Intel Core i3-8350K, a quad core CPU without hyperthreading. This is far from ideal for your setup.

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