Well, as the OP already mentioned, it's not necessarily a drone workload issue. I have the same problem in my current game. All hubs between the starving dome and the farms have tons of idle drones, and the intermediary depots have 1 or 2 units of food MAX. The only depots that have reasonable amounts of food are the ones in range of the food producers.
"Oh, but you can use shuttles". No, you can't rely on them. They don't fly during dust storms and only activate when a service building calls for a certain resource.
"Oh, so you can use rovers". You could, but you would need to constantly move them around, which quickly becomes impractical and annoying in larger colonies. Why? Because the "set transport route" feature is almost useless. It's only real use is to gather metal from all around the map, not to create actual routes. You can't set how much resources you take at a time. You have to take everything until it's empty or the destination is full. And when that happens the rover just stops - and it can take a long time until you realize it stopped. Also you can't set up a route for more than one resource, nor can you make the rover pick and unload different resources in both it's starting point and destination. All these things that you would expect from a transport route you cannot do without managing it closely and constantly. Transport rovers really need some love from the devs here.
Yes, the drone AI needs tweaking, but I think the best solution for this is just to allow more customization in the depots. It's not enough just to build depots and forbid certain resources from universal depots. Depots could also have priorities and/or the possibility to set minimum/maximum amounts of each resource and/or the possibility of linking depots, effectively creating a supply chain that your drones would try to fulfill. One could argue that this adds more micromanagement to the game, but I think it would actually have the opposite effect, since the player would just make the initial setup and then forget about it.