The PSP sets up the memory controller before the x86 cores are started. It's not implausible that the PSP has some sort of cache as RAM stage, but that's before Coreboot starts.
With a fair bit of effort; notably, DMA and IOMMUs probably won't work, and most modern devices don't really support PIO. You might be able to boot a really simple environment that runs out of an initramfs though. It's also entirely not obvious to what degree you can use the paging system.
It'd likely be a substantial effort to port Linux.
Sure, I would not expect to do any advanced IO. But right, probably there is no serial console really close to the CPU either, this is not a Raspberry PI. No idea what signals could be used for that.
Page tables I am not sure either. Could you still do it like in Linux 1.0? No idea what things looked like then, but I assume much less dedicated hardware support.
> probably there is no serial console really close to the CPU either
outb to 0x3f8 might work.
I know there were once versions of Linux that supported running without an MMU. Those versions, with very limited hardware support, might work in this mode.
Why? As a fun project with some old motherboard for example :)