The boot sequence is (at a high level): BIOS > Syslinux > KERNEL > SYSTEM
If you don't see the syslinux prompt on-screen, it's a BIOS or (very old) hardware problem. If you see the syslinux prompt but no LE boot splash it probably hangs early in the KERNEL stage of boot before the splash has been rendered. If you make /flash writeable (mount -o remount,rw /flash) and remove "quiet" from kernel boot params you'll see verbose boot output on screen and messages can sometimes help to pinpoint what the problem is (take an in-focus pic with a phone and share it).
I doubt there's too many users with c.2012 NUC hardware still in use, but in general there aren't reports of "my thing doesn't boot without a keyboard connected" and LE is intentionally designed to run on systems that don't have keyboards connected, so I'd be thinking a BIOS (might need updating) or hardware issues (PSU, coin-cell for BIOS memory, etc.) to be a more likely cause.