I watched the developer's video and no where did I find an explanation of why it has to be P2P, but more a matter of why the attempt was P2P.
The problem is the P2P model's complexity vs the standard server-to-client model. Between 6 computers on a client-to-server model you have 5 connections. One computer acts as the server. On the P2P model on the same set of computers we have a mind-boggling 15 connections. Now, we've all probably played other online games, and know how unreliable a single connection can be, much less 15 connections over only 6 computers!
Not only is it a drag on bandwidth, but each of those connections has a different ping. It's a simple fact that when gaming online, when you hit someone, you don't hit them when you think you do. With today's cables and distances from one seaboard to another, it will typically be over a 10th of a second before you hit someone if you have a great connection, a poor or slow connection and it's only worse, not better. Instead of fighting against such numbers, typical developers create subroutines that deal with "lag" so the game can keep progressing without the people playing it noticing.
If lag isn't dealt with in robust subroutines then you can expect to find a lot of bugs playing online, caused by an asynchronous gaming experience because of the number of connections and not having a definite server to resolve any timing conflicts. So, you would think it could either be resolved by the one doing the attack, or the one receiving the attack. I would assume by the problems described that the resolution of any attack is being made by the receiving computer. The problems seen have to do with an attack being placed, and then the receiving demigod changes position before the attack lands, therefore the attack can not proceed and the attacking demigod becomes confused (either standing there, attacking air, or walking around in circles).
P2P works fine in file sharing, where the actual speed and reliability of the connection doesn't matter, and if the connection fails a new one can be established.