Is Super Mario Bros. a kernel? Every NES, Game Boy, Sega Genesis, etc... game ran on the "bare metal" without anything resembling an OS (or even a BIOS, really).
Kernel means "core". Kernel-mode-only kernel is a contradiction. If you don't have separate parts, you can't have a core.
So if you write core software for an architecture without privsep you must not call it a kernel?
I'm not talking about privsep, just different modules. I gonna answer you with another question:
Did MS-DOS had a kernel?
Yes. We all lose here today.
It depends. Being downvoted by people that believes a memcpy() is a kernel, is victory.