Beavers pick the closest drink, food, fuel according to where they are when they have the need to replenish, not when they're done with their current job.

The trick is that they should never have to haul long distances in general (not necessarily the job hauling, but just in general should not be traveling long distances).

I don't think that is true.
They do check for the nearest food/water/etc when they are done with the current task, however, I have witnessed, that at least BOTS seem to check distance not by PATH distance, but simple MAP distance.
I have seen bots run half around the map for biofuel, because there was a biofuel tank only a few spaces away, but not directly connected to where they were currently standing, running (or rather slogging) past _several_ biofuel tanks afterwards.

Many times they walk past the convenient barrel of water. Pathing doesn't always work.