"The Stormlands: They would declare Gendry Baratheon their King and that would be that."
Now, how would that be that? There's no reason to put some street kid from King's Landing in charge.
Oh, you say he's Baratheon's son? Prove it. If that's the case he should have been brought up by some loyal retainer of Baratheon's, who the other lords would believe. But he wasn't.
And even if he was Baratheon's son, he's a bastard, so that's irrelevant. The last thing the lords want is bastards showing up and making claims, embarrassing the family. It's also illegal. Bastards do not inherit.
He has no lord vouching for him. No soldiers, no gold, no allied houses outside the Stormlands. The only "proof" is that the crazy foreign woman who burned the capitol wanted to put him in charge as her lackey. That works to his disadvantage. Very much.
House Baratheon is gone and there are others who want the seat. That's how it works.
As for King's Landing, it would never become a burned landmark. It's not on an island like Valyria was - which only remained uninhabited because of the magic-caused diseases that affect people on the island. King's Landing is on the mainland and has a great harbor, the best harbor on the continent, in a central location. You don't leave great harbors just because the buildings were burned. You build new buildings.