@ned No... thats not it.
Historians are split on if **any** kingdom existed, either a uniqued one or split.. There are two versions of the scrolls which were eventually the book of samual. You have teh monarchical version and the non-monarchial version.
In the non-monarchical version of the scroll (not the one that made it into the bible) Saul/Samual refused the call to become kind and no such kingdoms were every formed at all. In the monarchical version the kingdoms are part of the story and he accepts the call to become king and forms the kingdom.
The issue is that the monarchical is litered with enormous errors that many are anachronisms.. things that could only be known if the writer was writing many years AFTER the claimed existance of the kingdoms (things that didnt exist int he area yet were written as if they did). This lays pretty clear evidence that of the two the monarchial version is unlikely to be historic suggesting the other copy is more like to be fact.
The idea that the **only** piece of physical evidence is a stone sitting int he sand with the phrase "house of david on it" that is 60 to 180 years after the existence of such a mythical nation lends little if any evidence to the story and simply suggests someone who believed int he mythos wrote the phrase in a piece of stone at one point for some reason.