[2023] Multiplayer Game Can't Start While I'm Present.

I have no idea what could cause that, I'm sorry! A weird one.

Has this game always been like that, or it was okay up to some point?

If you are all on Steam you'll be using Steam MP. In our launcher's Game Settings menu select Cross Store MP (all players will need to do this) - do you get the same problem then?  

What if you host a game, does that work normally?

Do check that all Steam ports are open (NOT forwarded, open) in all your firewalls:

Steam Support :: Required Ports for Steam

support.steampowered.com