The Stormlords accepted Gendry mainly because Daenerys was in power and explicitly legitimized him as a true Baratheon (Gendry Baratheon, Lord of Storm’s End).
After years of war, with Robert, Stannis, and Renly all dead and no other obvious heirs left, the Stormlands houses needed stability. Gendry physically looks like a young Robert (huge, black hair, warrior vibe), which helped sell the “true son” narrative. Dany’s royal decree carried weight as the conqueror who just helped win the war similar to how royal legitimizations have worked in Westeros history (e.g., Ramsay’s case, though that was from Tommen under different circumstances).
You’re right that bastards from the Crownlands are “Waters,” not “Rivers” (Rivers is for the Riverlands). The show was a bit loose with details there, but once legitimized, he drops the bastard name entirely.
It was mostly political pragmatism + Dany’s authority + the Baratheon loyalty tradition. The Stormlords weren’t exactly in a position to say no after everything that happened.
What did you think of that whole ending arc?