At a customer, we were customizing and locking down the Start menu using StartLayout.xml and Group Policy at some locked down computers.
The strange thing was, that desktop apps (W32/x86 programs), that were supposed to get pinned, didn’t appear in Start. But Modern apps appeared fine.
Every other requirement was taken care of.
It turns out, that two group policy settings were the root cause.
If either of these two Group Policy settings are enabled, addition of pinned desktop apps to Start will be blocked:
Remove common program groups from Start Menu
Remove user's folders from the Start Menu
Both of them are placed in Administrative Templates\Start Menu and Taskbar
Once these Group Policy settings are removed, the desktop apps are pinned.