Also, you must boot the M1 Mac by holding the power button down until you get the menu. Then at the menu hold Command + D until it begins to boot (or reboot) to the Apple Diagnostics (no idea why it cannot have its own menu option or why they have to stray from the Intel keys). You do need a good Internet connection for the Apple Diagnostics.
FYI, some Apple Silicon Macs may not have a fan (I forget which model does not have a fan, but I think the Pros do have a fan). Also it would not surprise me that a macOS update may have broken a third party app like MacsFanControl even if the laptop is nomally supported. I'm not sure whether it is possible to adjust the fan speed of Apple Silicon Macs. I don't believe I have tried MacsFanControl on a Silicon Mac yet.
Is the Mac being managed by a school or business? If so, then maybe the Mac is configured to to restrict special boot options.
It is possible you may need to "Revive" the firmware or perhaps even "Restore" the firmware (the latter is data destructive). I would make sure to have a good backup before even attempting the "Revive" option. The Apple documentation doesn't mention anything about the Apple Diagnostics, but the diagnostics are similar to recovery mode as it leverages the same framework and it is sort of a boot issue. This is one thing Apple will likely do themselves if they examine the laptop. I've seen a lot of odd quirky behaviour with all of the Apple USB-C Macs.
Revive or restore a Mac with Apple silicon using Apple Configurator - Apple Support
You can always take the laptop to an Apple Store or an Apple Authorized Service Provider to be examined so they can run their service diagnostics.
