(disclaimer: I only play Ironmode so my suggestions may or may not only reflect particular needs of that game play; disclaimer 2: I am not a game developer so my suggestions most likely don't take into account the complexities of implementing the suggestions below into the game)
I think it would be a lot better if one could obtain log spirits as one practices woodcutting, in a similar fashion as how one obtains ore spirits as one practices mining. The intermediate steps of needing to unload bird nests at the bank, and then filling the wood box with any awarded spirits appears to add unnecessary friction, which goes against the intended convenience of adding the wood box/wood spirits into the game.
Having that said, if the suggestion above was incorporated, then wood spirits could be either (i) automatically stored in the wood box (i.e. as they are acquired from the woodcut practice) OR (ii) they could go directly in the player's inventory (to be subsequently added onto the wood box manually). I think option (ii) might be more practical from a logistics standpoint, as you will see below.
If option (i) above was incorporated, then I think it would be nicer if the empty function could be customizable; i.e. one could check boxes in order to choose which items are unloaded in the bank, in similar fashion as how seedicide and bone crusher are customizable. For example, if I am on an elder log grind and I have not yet reached the elder log spirit cap of 240, I might want to toggle the empty function in such way that it unloads all the contents of my wood box except for any elder log spirits I may have stored in it. Further, if I haven't reached the spirit cap of 240 for other spirit tiers, I might not want to unload those spirits in the bank either (to save bankspace).