Need to replug my Magic Keyboard after each shutdown
On my iMac running Catalina I can enter my password after reboot using the bluetooth Magic Keyboard without connecting it via cable. Although you run a different version of MacOS on a different machine it seems to me that this is rather a bug than a feature. Does it also appear after reboot or only after complete shutdown?

I have two ideas you can try to fix the problem: First, reset your SMC. If this doesn't solve the problem reset your bluetooth controller.

If both things don't work, connect the keyboard to another Mac to see if the problem is the same. If it is, I would order a new Magic Keyboard or go to an Apple store.