Actually, the need to establish a Christian government in what is now Turkey was well understood by at least one of the Allied powers: Russia. Unfortunately, the German use of Lenin as a weapon of mass destruction had removed Tsar Nicholas II from power by the time it mattered.
There had been an agreement to give Constantinople and the region around it to Russia, and it was The Holy Royal Passion-Bearer Nicholas’s intention to reestablish a Christian Empire with its capital at Constantinople.
The opportunity was lost again when the Western powers refused to support the Greeks in the Second Greco-Turkish War.
The fall of the Tsar was a great calamity. In Orthodox Christian thought, the Christian claimants to the throne of Caesar — the Emperors at Rome and Constantinople and the Tsars of Russia after them — are “the one who restrains” who must be set aside before “the man of lawlessness” (the Antichrist) is revealed. Actually in a few years from 1918 to 1922 every conceivable claimant to the throne of Caesar — the Orthodox Tsar, the Lutheran Kaiser, and the Muslim Sultan (the Ottomans when it suited them regarded themselves as Muslim Roman Emperors) were overthrown. We are living in the end-times, the only question is how long they will last.
Orthodox Russia in fact prepared for WWI with the aim to take Constantinople
Any reading of Muscovite history shows that it has had 2 obsessions: the first from 1240ish until 1791 was 'the gathering of the Rus' when Muscowy believed that it was divinely ordained to gather together the various Rusyn peoples (remember that the Tsar was crowned "Tsar of ALL THE Russias")
The next obsession from 1490s onwards was "the gathering of the Orthodox" -- a divine mission to re-capture Byzantium
That latter mission fueled Russia through countless wars with the Ottomans and the Iranis
In fact it so permeated them that Tukachevsky, an avowed anti-Christian felt that Russia needed to capture Istanbul...