There is an interesting alternate history book by Antoine Reverchon on the war of 1870 (La France pouvait-elle gagner en 1870? - Could France win in 1870?), which I read while searching for material for my 1870 TL, with several scenarios over several pods involving breaking the siege of Paris or even avoiding it. Aside of some implausible assumptions on political ground, the author raises very interesting tactical and strategical points.
Though I knew of MacMahon wish to withdraw to Paris through Schnerb biography of Eugène Rouher (
Robert Schnerb,
Rouher et le Second Empire, Paris, A. Colin, 1949), I wasn't aware of an Orléans plan. Honnestly, I thought of it too while playing the armchair general, but the more I looked, the more I saw the Paris defense is the better option.
The danger of Paris being cut off is overexagerated.
With the Army of the Rhine escaping Metz and getting reinforced by the corps raised at Chalons, plus further mobilized ones through the autumn (French manpower was still able to provide for new armies as the Republicans showed ITTL). Meanwhile, the Germans have to keep at least two armies besieging the fortresses of Lorraine and Alsace and keep the supply lines safe against raids or guerilla by the Corps Francs : Metz and Strasbourg, but also Toul, Nancy, Belfort, Mezières, Bitche... (that last one is real, no joke
https://fr.wikipedia.org/wiki/Siège_de_Bitche_(1870-1871) ).
If my memory is still correct (it has been a long time), that keep the Ist and IInd armies in Lorraine and Alsace and leave only the IIIrd and IVth armies to pursue the French and attack Paris.
In that situation, the Germans would likely be themselves outnumbered if not in parity with French forces before Paris.
Plus, while the French army is sitting on the biggest rail and supply hub of the country, with plenty of ammo at hand, the IIIrd and IVth German armies have to supply through a long, arduous and threatened supply route through a still contested territory (as long as the fortresses hold and the guerilla keep blowing up bridges and tunnels), not forgetting that French cavalry would have likely torn down every single railroad track and bridges between Verdun and Paris while retreating.
So, when it comes to cutting Paris, the Germans can't really go away from Paris, either north or south towards the Loire, without having the concentrated French army in their back.
Dividing is an option, but not a good one either.
They could keep an army to fix the French army at Paris and send another away to cut the lines to Paris, but that's forgetting the weakness of the German army there.
First, with the whole French army concentrated there, they don't have the numbers to besiege Paris, then, the French army has the benefit of interior lines and extensive railroads, so they can dispatch easily and very quickly corps from a section of the front and the battlefield to another. Moreso, to encircle or cut Paris, the Germans would have to cross the Seine river at some point, which would divide their forces and leave them exposed to being crushed in detail by a defender who has as we said the ability to ferry troops and whole corps quickly across the whole front and including across the Seine river.
Minding that, I don't think the Germans would outright seek to assault or besiege Paris. Before the Ist and IInd armies in Alsace and Lorraine are freed up, they can just try to provoke the French army into battle in hope of destroying it in the field.
But at this point, all the French need to do is waiting and keeping their army concentrated there in such a way, while engaging in peripheral action, launching deep raids from Belfort and Mezières towards Lorraine, harass German lines, make the fortresses resistance the longest possible, and wait for winter to do its work.
At the time, France was maybe outclassed tactically by Prussians on the battlefield, but strategically, it had a way better capacity to sustain attrition than the Germans had. They dominated the sea and had an excellent financial standing.
Meanwhile, the Germans in 1870 as later IOTL showed bad at sustaining attrition, their mentality being built on the idea of quick and decisive victories.
After, my idea of a more successfull French performance in that war was to have Napoléon III successfully lobying London and the European powers (better French performance would have made Austro-Hungarian not refraining from openly supporting France, diplomatically I mind, and after the French withdrawal from Rome ordered by Napoléon III had paved the way for Italian involvement) for mediation and for the Germans, badly affected by attrition, to accept it, and conclude the whole war by a status quo ante bellum.
EDIT: On the interior lines, you have this railroad to speak of:
https://en.wikipedia.org/wiki/Chemin_de_fer_de_Petite_Ceinture