1440p, 165Hz you could go either Intel or AMD without much difference.
AMD will be a little lighter on the wallet, generally.... but you could spend as little as ~$300, assuming your current DDR4 is of decent speed:
6c/12 AMD
PCPartPicker Part List
CPU: AMD Ryzen 5 3600 3.6 GHz 6-Core Processor ($199.99 @ Amazon)
Motherboard: Gigabyte B550M DS3H Micro ATX AM4 Motherboard ($94.99 @ B&H)
Total: $294.98
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-05 18:03 EDT-0400
8c/16t AMD
PCPartPicker Part List
CPU: AMD Ryzen 7 3800X 3.9 GHz 8-Core Processor ($339.99 @ Amazon)
Motherboard: MSI MPG X570 GAMING PLUS ATX AM4 Motherboard ($159.99 @ B&H)
Total: $499.98
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-05 18:04 EDT-0400
(you could drop to a 3700X here and save ~$40 and trade off a bit in clock speed)
6c/12t Intel:
PCPartPicker Part List
CPU: Intel Core i5-10600K 4.1 GHz 6-Core Processor ($274.89 @ Amazon)
Motherboard: ASRock Z490M Pro4 Micro ATX LGA1200 Motherboard ($134.99 @ Newegg)
Total: $409.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-05 18:07 EDT-0400
8/16t Intel, last 'gen', but a great chip:
PCPartPicker Part List
CPU: Intel Core i9-9900K 3.6 GHz 8-Core Processor ($399.99 @ Adorama)
Motherboard: ASRock Z390 STEEL LEGEND ATX LGA1151 Motherboard ($135.99 @ Newegg)
Total: $535.98
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-05 18:07 EDT-0400
Up to the full ~$600
10c/20t Intel:
PCPartPicker Part List
CPU: Intel Core i9-10850K 3.6 GHz 10-Core Processor ($486.83 @ Newegg)
Motherboard: ASRock Z490 Phantom Gaming 4 ATX LGA1200 Motherboard ($129.99 @ Newegg)
Total: $616.82
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-05 18:05 EDT-0400
For strictly gaming, either 6c/12t option should be plenty - with the slight edge to an overclocked Intel i5-10600K over the Ryzen 5 3600.
Of course, this assumes your current DDR4 is sufficiently fast. I'd recommend pairing at least 3000MHz with any of the above combos... preferably 3200-3600MHz.
What PSU do you have, specifically? With a 1080, you're looking at at least ~400W peak power draw on any of the Intel combos when overclocked