(I will focus mainly on France and try to explain some of its foreing policies in Central and Eastern Europe after WWI)

The short story would be that this stance was greatly affected by the views of basically all of the newly created states in Central and Eastern Europe and the creation of the so-called Little Entente meant that Habsburg restoration was practically impossible anyway - save for Hungary and even there the situation was complicated as I've written in my previous contribution, the former Emperor Charles was very unpopular as was the idea of a Habsburg restoration in Hungary with Czechoslovakia and her foreign minister Edvard Beneš emerging as the leaders of the anti-Habsburg movement in Central and Eastern Europe. The primary concern for these countries was that the ascend of a Habsburg on the Hungarian throne would prompt a revisionist invasion of their countries by Hungary and while they could likely defeat Hungary, it would become much more complicated were the Hungarians able to enlist a help of a Great Power. It was simply a risk they were not willing to take.

The long story however is that the view from Paris on the matter was rather complicated. Pro-Hungarian French diplomat and Secretary General of the Foreign Ministry Maurice Paléologue under the prime minister Alexandre Millerand, backed by business leaders and right-wing politicians, in 1920 came with a proposal, considering that at that time Poland was losing the war with the Soviet Union and the Czechoslovaks and Poles could not find any common ground for a potential joint action against the Soviets, for the so-called 'Danubian Union' - the exact form this union should take was still in question, but basically it was supposed to be a confederation of states, primarily economic, in Central and Eastern that would help counter the Soviets and be a beacon of French political influence which necessitated some form of a cooperation with Hungary that would be its key member. This was also in the interests of the French businessmen, it would open up the Hungarian market for French companies and it would make it easier to negotiate deals than dealing with each of the nations separately. For this goal France even said that it would consider (though never promise) a border revision in favour of Hungary, provided it signed the Trianon treaty first that is, or at least further mediate the dialogue about this between Hungary and its neighbours especially reagarding the borders with Czechoslovakia.

Anyways, the project fell through, partially because the Polish won the battle at Warsaw, and drove the Soviets back which made the project less important and partially because Prague grew suspicious of these negotiations, even when they were assured by Paris that they were strictly economic, and Beneš started building a bloc of countries (without informing Paris beforehand by the way), which would however only fully materialise in 1921 after Charles' first attempt at a coup, that would later become the Little Entente aimed against Hungary and the Habsburg restoration. The creation of this bloc actually angered Paris because it effectivelly sabotaged the French project, it was also concerned that the creation of this alliance would lessen the French influence in the region.

The reason I am writing this is that since the failure of the project, the French diplomacy started toying with the idea of putting a Habsburg on the Hungarian throne in 1921 in order to not only attempt to resurrect it but especially Paris thought that such a move could disassociate the Republic of Austria from Germany. However this was torpedoed ironically enough by Charles himself with his first attempt at the Hungarian throne during which he by the way claimed that he had an approval from the Entente including the French prime minister at that time Aristide Briand (and there is a good chance there was indeed some sort of an encouragement by the French). Czechoslovakia protested against this attempt at Habsburg restoration with Beneš managing to rally the neighbouring states against it, and the French government denied any involvement in the matter. This affair would also serve as the catalyst for the full creation of the Little Entente, especially it convinced Romanian Foreign Minister Take Ionescu, who was somewhat sceptical towards the idea initially because this Czechoslovak-led project could damage Romania's relations with Poland with which it shared concerns about the Soviet Union unlike Yugoslavia and Czechoslovakia, to join the project.

In any case the creation of the Little Entente de facto ended any hope for a Habsburg restoration (and France realized this as well), these countries were willing to militarily intervene in Hungary were this to happen. This became apparent after Charles' second attempt at the Hungarian throne when those countries mobilized their armies and Beneš demanded formal exclusion of Habsburgs from the Hungarian throne to which Horthy and the Hungarian politicians more or less complied, Charles was expelled and the Pragmatic Sanction from 1723 was repealed, though the prime minister of Hungary István Bethlen did not explicitely exclude the possibility of a Habsburg monarch on the throne for the sake of a compromise between the divided political factions in Hungary.

Source: WANDYCZ, Piotr Stefan: France and Her Eastern Allies, 1919-1925, 1962