The IPC gains of 3000 series chips would be beneficial in gaming.
However, 3600 + X570 isn;t your only option.
The cheapest I can see a 2600 + B450 is ~$205:
PCPartPicker Part List
CPU: AMD - Ryzen 5 2600 3.4 GHz 6-Core Processor ($139.89 @ OutletPC)
Motherboard: ASRock - B450M-HDV R4.0 Micro ATX AM4 Motherboard ($64.89 @ OutletPC)
Total: $204.78
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-07-12 19:28 EDT-0400
Vs 3600 + X570, cheapest ~$360
PCPartPicker Part List
CPU: AMD - Ryzen 5 3600 3.6 GHz 6-Core Processor ($199.00 @ B&H)
Motherboard: ASRock - X570 Pro4 ATX AM4 Motherboard ($159.99 @ Newegg)
Total: $358.99
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-07-12 19:28 EDT-0400
Or, a meet in the middle, 3600 + B450 Tomahawk (or a few select others), ~$315:
PCPartPicker Part List
CPU: AMD - Ryzen 5 3600 3.6 GHz 6-Core Processor ($199.00 @ B&H)
Motherboard: MSI - B450 TOMAHAWK ATX AM4 Motherboard ($114.89 @ OutletPC)
Total: $313.89
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-07-12 19:30 EDT-0400
That MSI board supports flashing the BIOS without a compatible CPU, as do a few other MSI boards.
Ok, it still costs ~$100 more than a 2600 (or 'saves' you $100 vs the 3600 + X570), but gets you the IPC gains of the 3600 which, for gaming, would be a good choice IMO.