I watched yesterday Alec Ryrie's talk on almost precisely this topic, which leaves me in the kind of awkward position of being strongly influenced by an youtube video, even though I think a pretty good one. In short, I am going to argue, that the Catholic church, as well as protestants in 1910 were in the process of meeting the challenge posed by the success of the scientific method (as understood at the time) and that such an argument is plausible but in a quite different context than early 21st century "culture wars" would suggest.
The believe that God created the world quite recently goes at least back to Gregory of Tours in the 6th century and in medival times was the dominant view. It came under pressure in early modern times due to Egyptian king lists and later, and more challenging, due to geology. However before the 19th century the church could reconcile science and doctrine, to some extend because the scientific evidence in doctrinally interesting areas was not yet strong.
That changed with the further development of geology and paleontology in the 19th century, when scientists started to collect more and better evidence. Darwin then published On the Origin of Species in 1859, and though it was not the only work that proposed some kind of evolutionary theory at the time, it became synonymous with evolution and it is at odds with the special creation of man.
There was a third important challenge to doctrine in the second half of the 19th century, "higher criticism," the idea that one should engage with the bible in the same way as one engages with other sources, for example trying to figure out in which order the gospels were written and by whom. Altholz (1975) argues that this is the most important challenge of modernity to doctrine, since it does not just impact some parts of the content of the bible, but how a Christian should engage with scripture.
For protestants, different churches found different answers which lie in a spectrum between a traditionalist view that affirms primacy of spiritual knowledge and a liberal view of trying to accommodate scientific views into theology. The catholic church in the late 19th and early 20th century also struggled with these challenges and in a way tried to split the difference by trying to set up a dialectic between the scientific and spiritual realm. Pope Leo XIII publishes in 1893 the encyclical Providentissimus Deus:
- […] There can never, indeed, be any real discrepancy between the theologian and the physicist, as long as each confines himself within his own lines, and both are careful, as St. Augustine warns us, "not to make rash assertions, or to assert what is not known as known." If dissension should arise between them, here is the rule also laid down by St. Augustine, for the theologian: "Whatever they can really demonstrate to be true of physical nature, we must show to be capable of reconciliation with our Scriptures; and whatever they assert in their treatises which is contrary to these Scriptures of ours, that is to Catholic faith, we must either prove it as well as we can to be entirely false, or at all events we must, without the smallest hesitation, believe it to be so [...]
That should however not be read as physics can disprove the bible, it continues a little later:
[...] And if writers on physics travel outside the boundaries of their own branch, and carry their erroneous teaching into the domain of philosophy, let them be handed over to philosophers [...]
The Catholic church goes so far as to excommunicate some scholars and declare in 1907 with the encyclical Pascendi dominici modernism, that is treating everything including spiritual matters like interpretation of the bible as a scientific question, as heretical and mandates in 1910 an Oath against Modernism:
[...] I also condemn and reject the opinion of those who say that a well-educated Christian assumes a dual personality-that of a believer and at the same time of a historian, as if it were permissible for a historian to hold things that contradict the faith of the believer, or to establish premises which, provided there be no direct denial of dogmas, would lead to the conclusion that dogmas are either false or doubtful. Likewise, I reject that method of judging and interpreting Sacred Scripture which, departing from the tradition of the Church, the analogy of faith, and the norms of the Apostolic See, embraces the misrepresentations of the rationalists and with no prudence or restraint adopts textual criticism as the one and supreme norm. [...]
To return to your question, the Papal biblical commission writes what I would call an advisory "Concerning the historical nature of the first three chapters of Genesis" in 1909, where they are carefully avoiding the question if the age of the earth is a sacral or scientific matter:
Question VIII: Whether in that designation and distinction of six days, with which the account of the first chapter of Genesis deals, the word (dies) can be assumed either in its proper sense as a natural day, or in the improper sense of a certain space of time; and whether with regard to such a question there can be free disagreement among exegetes?
Reply: In the affirmative.
I would read that as evidence that there are Catholic theologians that believe the world was created in a week and theologians that believe the scientific account and therefore an argument between a paleontologist and a priest about the age of the world seems plausible. However I doubt that the position should be called young earth creationism, but it is rather an 19th century traditionalist affirmation of a primacy of spiritual over scientific knowledge, that is basically a more medival instead of an modern world view.