As sibling notes, macs can install themselves over the internet from firmware, which sounds like exactly what you want.
Macs can also boot in target disk mode, which does need another computer but also trivializes certain recovery tasks.
Several systems (Android, at least some PCs, I think Chromebooks) have either 2 root filesystems or a recovery system on their disk that can reinstall the main OS. This is weaker but still useful.
You can flash iPXE to a network card and use that to boot a PC over the internet kind of like a mac with more steps.
Coreboot can take arbitrary payloads; if your device has enough flash you could easily include GRUB, FreeDOS, and a BASIC/LISP/etc.
A traditional BIOS can also boot an arbitrary payload. You can hijack INT 0x18 (boot to BASIC) to do whatever you want.
A lot of clone BIOSes just had to branch to a "please insert bootable media" or "No ROM BASIC - System Halted" message, but you could fairly easily inject something much like this.
I've modified the old "Anonymous XT clone" BIOS to support a stub that copies a chunk of ROM into RAM and then executes it.
This works reasonably well with some of the 512-byte "boot sector" tools I've tried, or with some minimal DOS .COM executables (i. e. an old Tiny Basic.) The copying step is a conservative approach because I suspect a lot of off the shelf software doesn't expect to be running inside ROM, and if it tries self-modifying code or simply writing to the same segment, all hell breaks loose.
For me, the limits were that, with an XT, you probably didn't want much more than 32-64k of ROM consuming precious address space, so unless you wanted to get gimmicky with banking in more memory or decompression, you're limited to maybe 20-40k of payload, which is a little small for "current" embeddable DOS products.
As sibling notes, macs can install themselves over the internet from firmware, which sounds like exactly what you want.
Macs can also boot in target disk mode, which does need another computer but also trivializes certain recovery tasks.
Several systems (Android, at least some PCs, I think Chromebooks) have either 2 root filesystems or a recovery system on their disk that can reinstall the main OS. This is weaker but still useful.
You can flash iPXE to a network card and use that to boot a PC over the internet kind of like a mac with more steps.
Coreboot can take arbitrary payloads; if your device has enough flash you could easily include GRUB, FreeDOS, and a BASIC/LISP/etc.