The original note, as written at the time
Once we figure out the battery issue, we might be able to maintain a fleet of drones dedicated to relay of communication channels. Because of their manoeuvrability, these drones will be able to be anywhere at all times, especially in places where they are most needed.
Ongoing development
Once the battery problem is solved, we could keep a fleet of drones dedicated to relaying communication channels — able to be anywhere at all times, and especially in the places where they are most needed.
The phrase worth keeping is the places where they are most needed. A relay drone is not a better tower. A tower is where the customers are on average; a drone can be where the need is today, which is the difference between a network built for density and a network built for events.
An event is exactly where communication fails: a collapsed building, a flooded road, a pilgrimage, a market on its busiest day, a border crossing at a crisis. In all of those the demand appears in hours and disappears in days, and the infrastructure is not there because it is sized for the average.
The battery is the whole constraint, as you say. A relay must stay up long enough to be worth launching, and a drone that must return to charge every twenty minutes is a gesture rather than a service. The version that works is a high-altitude loiterer with a tether, a solar skin, or a rotation of aircraft — and each of those is a different product.
The second constraint is spectrum and permission. A flying transmitter operates under rules that no country has settled, and a foreign aid drone transmitting across a border is a diplomatic incident with rotors.
The interesting design idea is the fleet rather than the vehicle: four aircraft, one of them always overhead, handing off so that the service never blinks. That is a scheduling problem, and it is solvable now.