To be honest, that question applies as much to the actual film - they never say why they're backing Shinzon's coup, other than the implication that the old praetor was too much of a peacenik for their liking.
The Romulans helping Shinzon's coup are also problematic. We never find out just how many there are other than two ships, and once it's apparent that their coup has failed, they're all ready to buddy-buddy up to Picard, who gladly accepts their offer even though throughout the entire film it showed that these Romulans wanted to invade Earth! Donatra makes some flimsy excuse about Shinzon not wanting to Invade Earth but to destroy it, but really, what's the difference?
My guess is they were using Shinzon as a figurehead for their takeover, and also as a convenient fall-guy in case he either screwed up or went rogue.
That's probably the one thing that makes sense in this entire silly plan of theirs, since that's essentially what happened. Although it still doesn't solve the inherent problem that Picard doesn't realize that these Romulans were the ones that wanted to invade the Federation in the first place.
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