You are misunderstanding French political factions at this time. Within the Monarchist faction there were two groups. The first favored the idea of the King but preferred someone reasonably competent (
i.e. the Comte de Paris). The second favored the legitimate descendent of Louis XIII (
i.e. in 1870 the Comte de Chambord grandson of Louis XV's grandson Charles X (brother of Louis XVI)). To be blunt both Charles X and his grandson were reactionary fools. However, to the "legitimists" this did not matter. I do not know the exact numbers but imagine that Republicans comprised 40% of the assembly, Orleanists (the first faction) controlled 30% and Legitimists (the second faction) control 30%. While the Legitimists wanted a king it MUST be a legitimate King. Therefore they will not vote for the Comte de Paris because he is not of the eldest line (he was descended from the
younger brother of the Louis XIV) Therefore while the elder line was still intact they would vote against him. In other words, using my hypothetical numbers above the vote would be 70% against the Comte de Paris (i.e. 40% of Republicans voting against and 30% of Legitimists voting against) The Monarchists (
i.e. both factions) thought they had a solution because the Comte de Chambord was elderly and did not have a son. When he died the House of Orleans
would become the eldest line and the Legitimists would accept the Comte de Paris. However, the Comte de Chambord messed up the deal by insisting on the Fleur de Lys rather than the tricolor (one example of the fact that he was a reactionary fool). The Orleanists would not support this, meaning you now have 70% voting against the Comte de Chambord (40% of Republicans and 30% of Orleanists). Therefore, a Republic was declared. By the time the Comte de Chambord died, and most legitimists accepted that the Comte de Paris should be king, Republicans held a majority . If you are interested, there is still a man styling himself Comte de Paris who is considered by French royalists (now a rather small faction) the King of the French (not King of France).
https://en.wikipedia.org/wiki/Jean,_Count_of_Paris