Why did the French political spectrum group into left/bellicose - right/pro-settlement in 1870.
There are both institutional factors and tactical factors.
Institutionally, the right (who's interest groups include the bourgeois, landowners, the prosperous and the commercial classes) has something to lose, while the left (representing workers and the proletariat) didn't. In a sense, the old Marxist quote that the workers have nothing to lose but their chains. If the French Army was destroyed, then there is nobody around to protect the bourgeois from an internal rebellion. Preserving the army, and preserving the army as a force capable of dealing with internal dissent instead of radicalized garde nationale elements in Paris, is absolutely vital to the established interests. The same logic happened in 1940; something had to be preserved of the state and of the army, in order for the reds to be kept at bay. It is important to note that the moderate Republicans, like Thiers, who were bourgeois themselves, were quite interested themselves in preserving their army to deal with the Paris Commune.
This plays out before the war to some extent too; from my recollections Clemenceau for example, when he heard of the declaration of war, thought that a defeat would undermine the Napoleonic regime and then enable the Republic to be formed; his mistake was that he also thought that popular mobilization would result in a subsequent victory against the Germans, when the situation compared to 1792 when this happened were different politically and militarily.
Tactically, the Republicans had their legitimacy from fighting against the Prussians. The Republicans were a coup against the Bonapartist regime after it lost all legitimacy following the disaster at Sedan. However, Bonapartist sentiment was certainly not gone overnight, and the threat of a restoration - from the Prussians supporting it, and here they kept Napoleon III technically as a "guest" after being captured, so that he could be used as a weapon against the French Republicans, from Bazaine who was quite possibly conspiring with the Prussians and certainly was quite interested in the political situation in France and keeping his army intact to bring about a peace agreement and then march back to have "saved" France and restore the Emperor, or from the provinces and countryside where the Napoleonic regime still enjoyed popularity unlike in the cities - would haunt the Republicans. There were also always the Orleanists or the Bourbons, and a monarchist restoration was certainly not an impossibility, and indeed came impressively close to happening post-war. By staying in the war, the French Republicans could continue without the need for elections that might return a majority against them and remove them from power.
Even more locally, there was also the power of the Parisian mob. The initial plan following the arrival of information at Sedan was for a compromise, moderate regime to be worked out. In the middle of negotiations over this, the mob broke into the meeting, and proclaimed the Republic, setting it on an initially very distinctly left wing course. Thus, the Republic, which was inherently left from the start, had even less right wing support.
Some moral elements come into play too. For the Parisian workers, it was a struggle between the Republic and foreign monarchist despotism coming to invade them, while their elites cowered and abandoned la patrie en danger and the wealthy and powerful preferred to treat with the enemy rather than with them, à la 1792. For the right, the narrative is much less convincing, and they had much more in common with the Prussians than the Parisian workers. Again, if we want to call to 1940, "better Hitler than Blum" had its share of chanters.
The Reparations demanded by Bismarck were not arbitrary figures .... they were exactly the same (per capita) as that imposed on Prussia by Napoleon after its loss at Jena-Auerstadt in October 1806. It was a case of "payback is a bitch".
Following the Austrian loss in 1866 at Koenigratz King William wanted to annex Bohemia and Moravia. Bismarck and the Crown Prince Fredrick softened his ire, couple with Franz-Joesph being astute enough to make a quick peace and avoid a drawn out post-loss drama. So much unlike the French Third Republic's intransigence .... Favre's early public pronouncement (19 September 1870) that "We will not surrender an inch of our territory, nor a stone of our fortresses!" was the beginning middle and end of his negotiations. It was left to a long drawn out process with American and British intermediators until January 30 th for an Armistice and February 26 1871 to reach a negotiated peace.
There does happen to be what the "quick peace" they're getting out of that is. The Austrians by accepting a quick peace get off with having lost their influence in the South German states and having to pay some small reparations - well, technically they lost Venetia to the Italians, but that wasn't to the Prussians per se. The French accepting a quick peace loses them directly two provinces that had been part of France for centuries, plus billions of Francs in reparations.... There is a slight difference between what the initial diplomatic opening offers were between France and Austria, and even if it turned out to be a mistake to prolong the war as it did, given that the French still had quite a lot of diplomatic and military strength left to them, it was neither illogical nor stupid that the French hoped that they might be able to moderate the peace terms.