I had the same problem of no tray icon this morning with my weekly back. There have been multiple reboots since the upgrade. It is a real thing.

Try a "repair install" to see if that will fix Teams. Mount the Win 11 ISO, run setup and select repair. You will not loose any data or app installs.

Good experience. Now put the HDD in a case and use it to store backup images of the OS on the SSD (you should have the OS and data in separate partitions).