The very fact that he spends so much time and effort obsessing about how Picard sees him is for me the selling point on the theory that he goes after Earth in order to impress. More specifically, he does it in order to impress Picard.
He may be a downtrodden slave and a victimized prisoner, but above all that, he is Picard, or almost so. Being a slave is just an excuse and ultimately a means; he's also a highly empowered soldier, and later a powerful rebellion leader. His life ain't so bad, not in Reman terms at least. But he's still only almost Picard. If Romulans did anything wrong, it was stopping him from being Picard. Jailing and forced labor was just how
they did it.
That part of the character works very well IMHO. It's no wonder he's all cardboard, when he was meant to be something and then allowed to be nothing. He has a single, extremely well founded goal in life, and he gradually works up to having the means to do something about it.
And that's where the movie really fails IMHO. Going after Earth is a nice trick, just like sucking the life out of Picard is. He could do both without needing any reason other than "this is how I get Picard; this is how I get to be Picard". But his plotting should culminate in that. At one point or another, he should attempt to usurp Picard's powers and position, not just his blood.
As the movie goes, we don't really learn whether Shinzon cared about Earth one way or another. The entire attack could have been pretense, a ruse to force Picard to do... something, just like planting of B-4 and assassinating the Senate forced him to act in a certain way. But the movie is lacking in this respect: Shinzon's ultimate goal is never revealed.
I've ridden my hobby-horse around here before, but it's whinnying for another go, so... Replacing Data with B-4 is a fun if somewhat silly trick. The movie would conclude with much greater splendor if that one were mirrored by another: Shinzon finally getting what he wants, replacing Picard, becoming Picard. Had he been allowed to age from his Tom Hardy self to his Patrick Steward self, there could have been a heroic last struggle where it is left unclear which of the two Picards survives - all we know is that the survivor was the one who did the right thing in the end. Except that we're also shown that the one who didn't survive did the right thing, too, and heroically sacrificed himself...
Now there's a good send-off for the good old cast. When two heroes both survive and perish, how can it fail to be twice as good as ST2:TWoK?