They did have their act together, though. The Assembly in 1871 had 689 members, and about 350 of them sat on the right (that is, the extreme right, the right, and then right-center), compared to 339 Leftist Deputies. For further simplicity, Legitimists tended to be extreme right and right, while Orléanists were right-center. Conservative Republicans sat on the left-center. So Monarchists dominated that initial assembly. They had a slim majority that would've allowed them to restore the monarchy. It wasn't because the Orléanists and Legitimists were fighting. Quite the opposite. The Count of Paris went to Frohsdorf and actually pledged to the cause of Chambord, and it was understood if Chambord was restored than the Count of Paris would be his heir. Chambord was the main issue: he dithered on the flag, and although in 1873 he declared that he rejected the "Phantom of the tithes, feudal rights, religious intolerance, persecution against our separated brethren… [and that he rejected] government of the priests, the dominance of the privileged classes," Thiers and Gambetta both succeeded in painting him as an ultra-reactionary. The Orléanists supported Chambord in 1871-1873 because they knew with his death (he was already in his 50s) that their candidate would become the next King of France.
I don't understand how Republicanism is only skin deep, considering by the next election in 1876 the monarchists (of all kinds, mind you) saw their share in the seats drop significantly, and the Assembly became dominated by the Republicans. The Monarchists had their shot in 1871-1873, and they missed it. The Monarchists won all their seats, not because the French people were geniunely monarchists, but because they ran on a pro-peace platform. Yes, the rural masses were conservative and the urban populace feared their votes would outswamp them. But I wouldn't say they were geniunely monarchist. After all, in the east, conservatism tended to manifest it's self not in electing a Legitimist deputy, but rather a conservative Republican.
Boulanger was also not so much a Monarchist as an opportunist. He courted all the Monarchists with promises of Restoration. But I'm fairly certain had he overthrown the Republic is would've merely placed himself at the head, with a government that did away with the weak executive and the strong Parliament. Boulangisme, after all (much like Gaullism) is a form of Bonapartism. He wasn't going to be a power behind the throne.