Between the revolution of 1789 and 1958 and France went through two empires, two monarchies and five republic. This was accompanied by revolts in 1848, the spasm of violence of the Paris Commune and the continuous and highly divisive culture war between the secular left and the Catholic right -- the most notorious example being the Dreyfus Affair. Why did the French do so badly in comparison to other European states?
I wouldn't say that France failed to create a stable political settlement as you put it. Looking first at the Republics post 1871, the Third Republic was very stable, it survived not only the Great War, but a devastating global depression, and was dissolved by the Nazi occupation, not internal destabilization. The Fourth yes did dissolve in what was essentially a military coup, but that doesn't reflect a french discontent with the Republican model, it was a "palace coup" led by generals who wanted to change the direction of french policy in Algeria, the state itself didn't change all that much (Alistair Horne in A Savage War for Peace: Algeria 1954-1962 goes into great detail about the caustic political situation in France during the war, and his chapter about the coup is very well researched). And despite the fact that he was put into power via a coup, De Gaulle's constitutional changes (essentially devolving more independent power to the presidency) were passed by plebiscite once he took office. It's tough to call the transition to the Fifth Republic a revolution, other than how it was installed, the state it created was not all that different.
Furthermore, I also think you're conflating a bit the "political turnover" of the nineteenth century with the twentieth. After the formation of the Third Republic, the idea of the democratic republic became immutable in french political culture, there was never any thought to going back to a monarchical or dictatorial model in 1945 or 1958, it was always going to be a Republic, there was simply disagreement as to what that republic would look like. The nineteenth century however was a continuous cultural conflict to define the state, but not between left and right but more between republican and aristocratic. François Furet makes a convincing argument that the French Revolution didn't really end until the formation of the Third Republic in 1871 because that was when the democratic institutions necessary for a Republic were founded (among other works, I would suggest checking out Interpreting the French Revolution). The nineteenth century was essentially an ongoing "argument" over to what degree should the state adopt republican measures. Despite how many french states there have been since 1789, it feels a little reductive to call it unstable