This keyboard is from around 2008 and the documentation I’ve readfound references Leopard (10.5.7)! Keep in mind this is BT 2.1 era. It is likely that this keyboard/BT standardisstandard is simply no longer supported by these newer Operating Systems. It’s not unheard of for Apple to simply drop support for obsolete products.
If your Mac connects to other Bluetooth devices, it’s technically working.
TestTry testing the keyboard with a different computer. If it works there, then it’s narrowed down to a compatibility between that keyboard and the OS.
What could be the incompatibility?
Based on when it was released, it could be the Bluetooth encryption method the keyboard employs is no longer supported. BT 2.1 and this keyboard (A1314; AA batteries, no recharging) were all released roughy around this time so the keyboard may not have the latest BT2BT 2.1 Specificationspecification that was available at the time. This will take more research to confirm. If it was an earlier BT spec, it may not be capable of encryption and the new OS might require it making them incompatible.