The Lumberjack rat has the area in the screenshot selected with the area staff, however it only goes for the really close trees, and only picks up one of each item before leaving the rest to despawn
Am I doing something wrong or is this just a bug, because I can't find anything about it online

which version are you playing on? I completely redid the lumberjack upgrade in 1.19.4/1.20 and havent been able to make this happen

I have the exact same problem on the latest 1.20.1 version

and I decided that a create mod tree farm would be better anyways