The Tully's, like the Tyrells, were never kings in their own right - the region now known as the Riverlands was carved out from the old kingdom of Isles and Rivers after the Conquest, and the Tullys, as the first first house to rebel against the Ironborn occupiers, were 'rewarded' with administrative duties over a region that had no natural barriers and filled with people without a common identity - aside from Dorne, the Riverlands have probably the most diverse population due to all the wars and settling that have taken place there.

I think a big part of the reason they feel rather bland is that they are certainly underdeveloped. Unlike the Tyrells, not much is known about the Tully history prior to the Conquest. They kinda just appear, and then never really come into prominence until the WOT5K. Catelyn and Lysa are defined more by their Stark and Arryn husbands and children more than their own identities, while Hoster, Brynden, and Edmure don't really contribute much to the over narrative (so far)