If Aegon takes King's Landing and smashes the Lannisters and Tyrells in power, I can't imagine too many people won't bend the knee to him.
I thought your theory was that Aegon can take KL because the Tyrell's are chasing down Ironborn in the Reach? There's no way Aegon can take KL if the Lannisters and Tyrells have their full strength there. He'd get smashed. The only way he takes KL is if it is essentially undefended because the Tyrell (particularly) and Lannister armies are elsewhere. He just doesn't have that many troops.
Who else are they going to bend the knee to? Stannis? He's a million miles away fighting the Boltons and no one likes him anyway.
Nobody. I mean, why bother bending the knee at all if the Crown cannot enforce its will? Perhaps a Kingsmoot to select a new King, and I don't think they'd take some complete unknown from overseas. Alternatively, I think Danys and her dragons would be a much more impressive option than some kid of questionable/
unproveable lineage leading an army of sellswords.
I stand by my projection that Randyll Tarly turns on the Tyrells and sides with Aegon in exchange for the promise of Highgarden when the war is over, and then Mace and his force side with Dany when she finally arrives. Not that Mace would have much choice at that point, as Margaery would be dead and any claim Mace's family had to the throne abolished.
Your idea of Tarly turning is pretty intriguing, but that still gives Aegon only an army of sellswords and Tarly. I do agree he might be able to seize KL. I just don't see what he can do after that. He'd have to just hope the rest of Westeros bends the knee to him, and I can't see why they'd bother. There's not even any basis to believe he is who he claims to be. And, Garlan Tyrell -- who is much more capable than his father -- is in the process of rallying the rest of the Reach to himself anyway, which is going to mean he's got a
lot of power gathered in one spot. Maybe Tarly turns on that, but damn, that's a risky play. The ironborn are pesky, but there really aren't that many of them, and Garlan should be able to smash them.
I still have no idea how Littlefinger and Sansa are going to play into all of this, though.
I don't either. But I'm not sure that will even be a major plot point other than just to resolve their storylines. Well....suppose that LF sees an opportunity to change his plot from marrying Sansa to Harry the Heir, to having Sansa marry Aegon? Maybe that would be the kind of thing that might get Aegon more support/legitimacy. Although Sansa is still suspected of killing Joffrey, so I'm not sure how much of a positive that really is.
Anyway, as I see it, Aegon is essentially trying to replicate what Aegon the Conqueror did, but without having any dragons. That's a damn tall order.