I'm a bit let down because "The Lodger" had potential to be an amazing study of the Doctor's psyche, but instead they just made a lazy stab at comedy.
I thought the episode was fun and witty and not lazy at all... but I agree a more serious, alternative version like you describe would have been interesting too.
Although there's the risk such a story could make him look like a complete imbecile, if he can't even clean or do the laundry. lol
For me the better option would be to take the lighthearted story we got, but work in a few more serious moments of self-reflection. Maybe just a scene of him on a park bench watching life slowly roll by, and realizing that he actually CAN enjoy this.