My assumption is: The wyse needs some special shutdown no one knows about
So there is no machine-specific shutdown procedure in LibreElec's kernel. I also assume that it's safe, but yet annoying, to power it down manually when the "Power Down" message appears.
I found a couple of other references, but no solution:
201965 – Power off / Shutdown hangs after echoing "Reboot: Power down" on DELL / WYSE Thin Client - AMD G-T56N
I'd start with ensuring the device has the latest/last available BIOS/firmware update installed, and seeing what if-anything can be configured in the BIOS for the device with respect to power states.
These devices do claim to support SUSE Linux (SLES 12) so any attempt to triage the process might require comparisons with that OS to see what state certain registers or things are being set to (or not) and upstream linux. How to diagnose power-state issues are way beyond my sphere of knowledge though. I did attempt to Google ACPI and references to Wyse but drew blanks. NB: Userspace shutdown scripts are not the problem or solution here. The issue (and any resolution) will be in lower-level code.