For strictly gaming, an Intel setup should outperform, due to the higher IPC, but Ryzen is a great option.
Ryzen does offer a solid "value" offering though - for the money, it's tough to beat.....but the 1600X is a tough sell.
The 1600 can OC to the same range, for a good $30 less while including a stock cooler.
The H60 is a fairly "basic" AIO, and I'd suggest either Air cooling, or a larger 240 AIO - for Ryzen, the stock cooler would be sufficient, as they're topping out ~4GHz, regardless of voltage.
The RX580 is a tough one to justify too, at the price. While prices are normalizing, a 6GB 1060 is ~$265, and even a 1070 is doable within the budget.
On the B350 chipset, the ATX boards don't make a lot of sense. Sure, a little extra expandability, but no SLI, CFx etc.
A couple of options I'd suggest:
Ryzen:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor ($193.88 @ OutletPC)
CPU Cooler: CRYORIG - H7 49.0 CFM CPU Cooler ($34.89 @ OutletPC)
Thermal Compound: Arctic Silver - 5 High-Density Polysynthetic Silver 3.5g Thermal Paste ($5.83 @ OutletPC)
Motherboard: MSI - B350M GAMING PRO Micro ATX AM4 Motherboard ($60.98 @ Newegg)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory ($156.34 @ Newegg Marketplace)
Storage: Samsung - 960 EVO 250GB M.2-2280 Solid State Drive ($117.49 @ OutletPC)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($45.69 @ OutletPC)
Video Card: EVGA - GeForce GTX 1070 8GB SC GAMING ACX 3.0 Black Edition Video Card ($399.99 @ B&H)
Case: Fractal Design - Focus G Mini (Black) MicroATX Mini Tower Case ($39.99 @ Newegg)
Power Supply: EVGA - SuperNOVA G2 550W 80+ Gold Certified Fully-Modular ATX Power Supply ($63.98 @ Newegg)
Total: $1119.06
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-10-24 13:57 EDT-0400
For Ryzen, the stock cooler should be sufficient for 3.9-4GHz. The H7 would likely need additional mounting hardware from Cryorig.
You could also go with a 240mm like the H100i V2 for this setup too, but probably not too necessary.
Intel:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i5-8600K 3.6GHz 6-Core Processor ($279.89 @ B&H)
CPU Cooler: Corsair - H100i v2 70.7 CFM Liquid CPU Cooler ($109.99 @ Amazon)
Thermal Compound: Arctic Silver - 5 High-Density Polysynthetic Silver 3.5g Thermal Paste ($5.83 @ OutletPC)
Motherboard: MSI - Z370-A PRO ATX LGA1151 Motherboard ($111.78 @ OutletPC)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory ($156.34 @ Newegg Marketplace)
Storage: Samsung - 960 EVO 250GB M.2-2280 Solid State Drive ($117.49 @ OutletPC)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($45.69 @ OutletPC)
Video Card: EVGA - GeForce GTX 1070 8GB SC GAMING ACX 3.0 Black Edition Video Card ($399.99 @ B&H)
Case: NZXT - S340 (Black/Red) ATX Mid Tower Case ($64.99 @ SuperBiiz)
Power Supply: EVGA - SuperNOVA G2 550W 80+ Gold Certified Fully-Modular ATX Power Supply ($63.98 @ Newegg)
Total: $1355.97
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-10-24 14:00 EDT-0400
While the Intel offering is over budget, it will outperform the Ryzen offering...... but both are powerhouse gaming rigs.
The 8600K is doing 5GHz for most people with decent cooling - and 5.3GHz+ has been reported while delidded (but YMMW, of course).
Both builds are mainly focussed on "best" gaming performance for the money (IMO), but your resolution will play a factor.
If you're playing @ 1080p/60Hz, then a 1070 is going to be overkill.
1440p would be the "sweet spot" for a 1070, and even lower settings @ 4K...... but if you want to do 4K, the budget likely isn't quite there yet - at least not without making some decent sacrifices.