Then you did not understand the principle of controlled residential ventilation. It does not extract air in the living room and blow it directly into the bathroom, but rather extracts the approx. 24-degree warm air in the living room and uses this energy to preheat the air for the bathroom. So it is distributed indirectly.
About chopping wood: yes, it's fun. I also like to do a few cubic meters sometimes. However, it also takes quite a bit of time. And together with the other disadvantages I have perceived, I also think that it is not "worth it" this way.
Probably, if I absolutely wanted to have a fire in the living room, I would choose a pellet stove.