The Lenovo ThinkStation Tiny Workstation series looks to be about the right form factor (can be mounted on the back of a monitor too). Only missing criteria is

The Lenovo ThinkStation Tiny Workstation series looks to be about the right form factor (can be mounted on the back of a monitor too). Only missing criteria is a dedicated GPU, but Iris Xe iGPU's outperform a GTX 260, so for only gaming it might be an OK compromise. They can be had cheaply second-hand. If you're willing to go slightly larger, Mini-ITX PCs are still quite small, but have the benefit of all desktop-class hardware and easier upgrades. There are too many options here to make a specific recommedation, but if you search for Mini ITX on /r/battlestations, you can find some inspiration.