What I meant is: supposing my router treats packets as it should, then opening the right ports would work, wouldn't it? And that wouldn't even require a NAT facilitator...
If your router is working as it should, opening the ports wouldn't even be required. The issue arises when the router can't/won't handle the numerous socket connections between you and the other players. Unfortunately, routers don't usually come with an option to adjust this number.
The second issue will be that even though you may have opened the necessary ports and done what you can to pave the way for the wonderful Demigod packets, that doesn't mean the other player have (thus the need still for a NAT facilitator). If someone in the game you're joining is not on a "friendly" network connection, it can cause problems for the group.