This answer only considers the nuances of habere, not a comparison between it and the possessive dative.
The possessive genitive is different; it functions mostly like the English genitive and is used to express things like "my dog" rather than "I have a dog".
The example of the pope actually makes a good example for habere.
The canonical announcement upon the selection of a new pope begins:
Annuntio vobis gaudium magnum: habemus Papam.
We have a pope, but we do not own him or rule him in any reasonable sense.
One could try to argue that this is post-classical, but no.
Please take a look at the entry for habere in Lewis & Short for a classical view.
It has a wide variety of uses, including clear ownership (having money) and clear lack thereof (having a brother).
The verb habere is very broad, and only rarely does it have the nuance of "to be the master [of]".
Assigning this meaning to it in general is not justified at all.