I can't get the desktop properties applet to run, no matter how much I
want it to.

Desk.cpl is the correct version, as is rundll32.exe.

Where should I continue to look?

I did a registry access trace using SysInternals' Regmon, both on the
computer that didn't work, and on the one that does.

In order to fix this, I had to import the following registry key:

Windows Registry Editor Version 5.00

[HKEY_CLASSES_ROOT\CLSID\{4c892621-6757-4fe0-ad8c-a6301be7fba2}]
@="Windows Settings Page"

[HKEY_CLASSES_ROOT\CLSID\{4c892621-6757-4fe0-ad8c-a6301be7fba2}\DefaultIcon]
@=hex(2):25,00,53,00,79,00,73,00,74,00,65,00,6d,00,52,00,6f,00,6f,00,74,00,25,\

00,5c,00,73,00,79,00,73,00,74,00,65,00,6d,00,33,00,32,00,5c,00,74,00,68,00,\

65,00,6d,00,65,00,75,00,69,00,2e,00,64,00,6c,00,6c,00,2c,00,37,00,00,00

[HKEY_CLASSES_ROOT\CLSID\{4c892621-6757-4fe0-ad8c-a6301be7fba2}\InProcServer32]
@=hex(2):25,00,53,00,79,00,73,00,74,00,65,00,6d,00,52,00,6f,00,6f,00,74,00,25,\

00,5c,00,73,00,79,00,73,00,74,00,65,00,6d,00,33,00,32,00,5c,00,74,00,68,00,\
65,00,6d,00,65,00,75,00,69,00,2e,00,64,00,6c,00,6c,00,00,00
"ThreadingModel"="Apartment"