I set up a lumberjack rat like that:

It is set to deposit in the chest and its home point is set to the cobblestone one block south of the torch. Radius is set to 8 with its center is the same as the home point. The second rat is setup exactly the same, except is it only gathering items.

The Lumberrat is trying to harvest the tree, but it is out of its range. If I push it in the right direction far enough, it will start to mine the tree, but it also tries to go back on top of the chest and cannot reach the tree again ...
I would expect the rat to move to the tree when trying to harvest, instead of its drop off location.

Eventually I managed to pushed the Lumberrat to the tree often enough for it to harvest the log. The tree collapses and the rats pick up a few items until they stop like this:

They do walk near items as expected, but not close enough to pick them up. If I give them a push they will pick it up and put it in the chest and then get stuck near items again. So they either need to move closer or have a bigger pickup range.

Oh, also Rats seem to not return to tier home point when idling.

Hope you can find the issue ...