In the context with debate about the recent North lore video (which I agree is rife with inaccuracies but nevertheless) it does bring up the question of how much influence and authority do you think the Barrow Kings actually wielded to make the claim of dominion over all first men, like something akin to the Fisher Kings or early Gardner Kings as potential comparisons?

Here’s how the WOIAF puts it:

“the Barrow Kings to their south, who styled themselves the Kings of the First Men and claimed supremacy over all First Men everywhere, even the Starks themselves.”

The use of the terms “styled” and “claimed” suggests their reach was larger than their grasp, that they held authority over the south-west of the North but not the whole. 

So more like the Fisher Kings or the Yronwoods.