Knowing nothing about hardware design myself, maybe you might have better luck with risc-v since it's open source, vs x86-64.
https://en.m.wikipedia.org/wiki/RISC-V
I know siFive has some boards that I think are for sale, that could run linux. you might be able to find datasheets for the risc-v implementation if you have interest in trying to design your own CPU/board etc.
Yeah. RISC-V has been the goal. But, I was hoping for something a bit simpler and easier to get a hold of as a bare chip (would be nice if a 6502 or similar could do it). That way, I'd be able to reduce the complexity and scope. I suppose I could use a soft core on an FPGA. Not confident on my gateware skills yet to be certain that I wouldn't waste time debugging something caused by my own faulty HDL or bitstream.
I have found low-cost Sophgo chips (SG2000 - RISC-V, mostly) but last time I checked they didn't have mainline support due to problems in the C910 cores not properly implementing IEEE-754 FPUs and the vector extension predating the RVV1.0 release. If the upcoming Sophgo SG2380 becomes similarly availabile, that might do it as they licensed cores from Si-Five that should be fully compliant with the ISA spec.