Earlier quoted context omitted.
Something like an STM32 Discovery board is a good option for recapturing the mid-90s magic. You can get a ~200-MHz Cortex-M4 or M7 with a few MB of flash, external SDRAM, and a display for less than $100. They have really basic hardware 2D accelerators. The on-chip peripherals are well-documented, but off-chip peripherals require some digging to figure out how to program correctly. You can debug with GDB surprisingly…
As there are several available, is there one in particular that you would suggest for this use case?
A low-res screen like this works well because the chip can't rescale its video output.
ST provides libraries for all the peripherals so it's pretty easy to jump in if you know C. I think microPython works on a lot of these boards, too.