I would go the road Klojum proposes. Get rid of the motherboard and graphics card and buy a motherboard with integrated CPU, for example with a intel J4125.
I have a asrock J3455, I bought it for 90€. You can't even buy a GT1030 for this price. I use it as a NAS, and a desktop pc running Manjaro. It's powered by a 40W power adapter. So it's cheaper and it uses a lot less power.
I wouldn't recommend it when you are a gamer, it's good to run emulation (even GC and Wii) and older games but not the newer titles.
If you buy a RPI on top of that, you aren't even on the price of a new graphics card.