I figured it out.
I use a script to configure various Windows settings after a new installation. I've used it for years without any issues for Windows 10 and 11. One of those settings was to delete web results in the Start Menu search (I've never understood that ability anyway, I don't want web results if I'm searching my PC). There must have been an update that's causing that setting to break things.
Long story short, enabling this setting broke the start menu. Once I removed it, everything was fine.
HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Search, BingSearchEnabled=0