I'm having this issue too but I wasn't able to solve it by doing the above steps. I'm on an iMac OS Monterey 12.3.1. I have the Launch BTT on startup ticked under Settings and tried with having BTT added and removed from the login items. It still doesn't launch on startup. Is there anything else I can try?
I confirm that this problem is still there on my Intel Mac running Monterey.
I have tried removing BTT from my startup items; deselecting and then reselecting the launch-on-startup option in BTT preferences; and fully rebooting my Mac, instead of just logging out and logging in again. But nothing works.