He had a genocidal impulse toward Picard. His world, his perfect life. He wanted to take it all away. "My life means nothing while you're still alive." he told Picard. He also wanted to show Picard that what "he" was capable of, show him that underneath all the nobility and Federation ideals, was a man capable of unspeakable evil. "Look in the mirror, see yourself!"
Again, the hatred of Picard specifically makes a certain level of sense. But the idea that his hatred of Picard would extend to wanting to exterminate all life on Earth before
wanting to exterminate all life on Romulus? It flatly contradicts everything else we learn about his beliefs and motivations as a former slave and Reman nationalist. The idea that he hates Picard just doesn't go far enough to explain this hatred of Humans; it's dramatically arbitrary because it is inconsistent with the level of anti-Romulan hatred previously ascribed to him, and with the level of curiosity and lack of hostile feelings previously established towards other Humans.
As for the Romulans, he had no need to kill them, since he'd already conquered them. A lowly human slave had risen to rule those who mistreated him so badly.
Not really. Worf said it himself -- a Praetor's power depends on the consent of the Romulan military, and coups are not uncommon on Romulus. If Shinzon wanted to hold on to his position as Praetor, that made him subject to the need to placate the Romulan admiralty, same as any political leader who needs the support of other elites.
So, again, the idea that he would suddenly decide to exterminate Earth just doesn't make sense given what we'd seen of his personality. He was curious about, and not hostile to, other Humans when he first met them; he was openly contemptuous of, and physically disgusted by the thought of intimacy with, Romulans; he'd been enslaved and oppressed by Romulans for decades; he was a Reman nationalist whose goal was the liberation of the Reman people from their Romulan oppressors; he was planning on moving against the Romulan admiralty; and he is a brilliant and cunning strategist who has used his talents to rise to power within the Star Empire in spite of his lowly status.
That he would endanger his project to liberate the Remans by suddenly developing a fixation on exterminating Earth is inconsistent with all of that. It contradicts the idea that he's a brilliant and cunning strategist, because the entire plan is bloody stupid; it means he goes and endangers his plans to save the Remans for whom he has fought so long and hard; it flies in the face of the idea that he's truly disgusted by the Romulans and hates them more than any other race; and it contradicts the interest and lack of hostility he felt towards other Humans earlier in the episode.
It flatly contradicts everything we learned about him earlier in the film, and it does so for no particular reason. He suddenly decides he hates Humans because he hates Picard for having privileges he didn't, and this change of mind occurs for no discernible reason. He just abandons everything that had defined him, for no reason, purely because the plot requires that Earth Must Be In Danger.
And Shinzon's own lack of connection to Earth robs the final act of the film of any dramatic weight. Shinzon has no real stake in Earth's fate, and therefore his hatred feels arbitrary and unmotivated. The action sequences therefore feel empty, instead of carrying any emotional weight; this is action for the sake of action rather than action taking place because of the full dramatic force of the antagonist driving the story.
The guy was nucking futs. Mentally ill. In his worldview it made sense.
1. I'm not convinced Shinzon was truly insane. 2. Even mentally ill characters need to have motivations that make sense. Shinzon's hatred of Earth files in the face of everything learned about him earlier in the film, and it saps the drama from the movie. It's dramatically arbitrary.
The film would have been much stronger had Shinzon wanted to exterminate Romulus.