> I haven't studied the protocol but that seems like it has some...obvious routing issues.

Yes indeed. I don't understand how the peer-to-peer relaying can possibly scale without some directed routing algorithm.

If my phone running Briar is literally handing off every as-yet-undelivered message to every other phone running Briar, we're going to pretty quickly become overwhelmed.

It'll have all the routing issues of a Wi-Fi mesh network, except at a vast scale. https://en.wikipedia.org/wiki/Wireless_ad_hoc_network#Briar