I think I'm in a similar situation - I remember my login item launching fine in earlier macOS 13 betas, but it seems like this has started happening around the RC versions of macOS 13. I've made sure to remove any duplicate copies of the app that the system could potentially search for login items within. Once I did that, it seemed like the system was now trying to launch the copy of my login item that resided directly within my product's build folder (i.e. .../Build/Products/Debug/<TeamID>.MyLoginItem.app). I found that deleting this copy caused the system to start using the one inside my app's Contents/Library/LoginItems directory. Not the most ideal solution but it worked for me. Using macOS 13.1 Beta (22C5033e).

In Terminal, run sftool resetbtm, authenticate if needed, and reboot. Also trying to reset the state of "Login Items Added by Apps", and I tried this, but said that the command wasn't found. Is that an issue in the first beta, perhaps?  If it helps, my context is different to the OP – I'm looking to move an login item previously installed using SMLoginItemSetEnabled to SMAppService, and I'd like to test resetting the state of the permission notification.