This isn't the case during Fire & Blood. The Lannisters had little to no role in the governing of the realm during the first 100 years of Targaryen rule. It grows later on, and by the end of the war sure the Lannisters are in a higher position than some families that will fall. But it hasn't gotten to that point yet and right now they're firmly behind the Hightowers, Velaryons, and Baratheons in the south.
No Lannister had ever sat on the small council to this point, but this seemed mutual. Aegon I and his successors had their favorites that stemmed from who initially supported the conquest. The Lannisters didn't seem in any particular rush to join in on the governance of early King's Landing, as it was a shytty city (literally because of the poor sewage) that paled in comparison to their own cities, and they would be second fiddle to poor weak houses like the Celtigars in the eyes of Targaryen kings.
So they were like fukk that. But even then, when shyt hit the fan with Maegor, Rhaena and Aegon the Uncrowned went running to the Lannisters for protection.
The Lannisters were the richest house with all of the resources of the west until Corlys came along....and even then, Corlys was new money that had ALL of his wealth tied to trade and his new towns Spicetown and Hull, while the Lannisters were old money that had reliable income regardless of how international trade was doing. Even during the first 100 years, they were the house that all houses that wanted to be rich and prestigious measured themselves against. And that was with them not having an interest in joining the small council.
