Generally the way P2P applications dealt with this is by assigning one peer as the "host" that would disperse one single source of truth among all the other peers (granted there are security problems here if the host peer is running a hacked or modded client). If the host dropped, the peers would use a consensus algorithm to pick a new host, exactly the same as how database clusters pick a new writer node if the old one dies.
It may be easier for devs to use the server model, but it's also more expensive - standing up centralized servers that can scale to millions of users is $$$. I think there would be a lot more P2P software competing with server-based software if NAT wasn't a thing, because software companies would be able to build extremely scalable software without having to pay for the hosting costs themselves. While Google Docs has to handle millions of users, if I were to "host" my own document I would only care about handling my friends - maybe two or three or up to a few dozen, but it would be very rare that I would host a document for 1000+ people. Developing an application that can handle a few dozen connections would be much simpler than one that handles millions, I believe.