Windows version 22H2 OS Build 19045.3693
1. correct - missing only on primary, always present on secondary
2. yes - any skin, including default skins like Diamond, Echo etc.
3. restarting explorer didn't resolve the issue (in fact slightly borked the taskbar and needed to reload the WB skin - still no start button)
4. although it did seem to reappear at one point after a reboot, after the next shutdown/start cycle it had gone again..