Magic Keyboard not working after Factory Reset

Since Big Sur, my Magic Keyboard was super laggy (aside from other numerous bugs...) and I read on the internet, that I could try to reset the keyboard to factory settings in order to solve the problem. I did that and afterwards my Device was not listed in the Bluetooth Devices list. I restarted my MacBook and it still didn't show up. I turned the keyboard on and off numerous times and also tried to plug in the usb cable into the keyboard for direct connection, which didn't work as well.

I already reset SCM, but that didn't help either. Do I just have to accept that the keyboard is broken now? Just bought it 3 month's ago...