Khan wanted the Genesis torpedo for unspecified reasons.
Khan wanted the Genesis Device because he knew it would make him immensely powerful. But his primary motivation, the thing that really
mattered to him, was revenge against Kirk. That's why he kept chasing the Enterprise
after acquiring the Device instead of hightailing it out of Dodge.
Nero lost his planet and went on a tirade against the Federation in an alternate timeline set 125 years in the past of the one he came from. Vulcan and Earth had nothing to do with Romulus blowing up in the Prime future but he attacks them anyway for his own narrow minded reasons.
And while his reasoning is irrational, his motivations at least make sense: He believes that Spock and the Federation could have stopped the Hobus supernova sooner and therefore blames them for a "sin of omission" by allowing Romulus to burn.
I agree that this is a dramatic failing of ST09. But to be fair, there's nothing in ST09 to indicate that he didn't
dispatch some sort of warning to the Romulan government, either.
Shinzon was the legitimate Praetor of the Romulan Star Empire.
The "legitimate" Praetor? DSN's "Inter Arma Enim Silent Leges" established that Praetors of the Romulan Star Empire assume office only after being confirmed by the Romulan Senate's Continuing Committee. Shinzon, on the other hand, assassinated the entire Senate and then appointed himself Praetor. He led a coup d'etat, not an election campaign. He was about as "legitimate" as Augusto Pinochet
or Idi Amin
You could argue he was merely a figure head and his Romulan collaborators intended to betray him but that's only speculation.
No, it's not speculation. The scene towards the end of the film where the Romulan Imperial Fleet admirals are talking to him make it very clear that they supported him because he promised to take a more aggressive stance against the Federation but that he actually had no interest in appeasing them. They were threatening to depose him if he didn't placate them.
And that's the issue -- he supposedly doesn't actually support the Romulan admirals' agenda, and he's supposedly obsessed with Picard, but we never get the sense of why he's obsessed with taking down Earth. Picard, that sort of makes sense -- resenting that someone with identical genes had so much privilege when he did not. And the need to harvest Picard's blood or whatever.
But the idea that Shinzon, who openly displays contempt for Romulans constantly throughout the film, who goes to the trouble of leading a Reman nationalist movement against Romulan control, would turn genocidal against anyone other than the Romulans? It's just dramatically arbitrary. It's the equivalent of doing a story where Spartacus decides he'd rather destroy Carthage.