Logitech has a support article that mentions problems customers may have with Bluetooth mice and keyboards not reconnecting under certain conditions (Brought to my attention from Reddit).
If your Bluetooth mouse or keyboard does not reconnect after a reboot at the login screen and only reconnects after the login, this might be related to FileVault encryption. When FileVault is enabled, Bluetooth mice and keyboards will only re-connect after login.
One of the potential solutions is to disable FileVault. I personally think this is a terrible idea. FileVault encryption is an important security and privacy feature. I don’t know if the bug lies with macOS or Logitech, but my potential solution is to ditch the product instead of disabling your hard disk encryption.