post WWI

They did! The original proposal in the Treaty of Sevres in 1920 was that Istanbul would be occupied by the international powers, and they did in fact occupy the city. That treaty was rejected by the Turkish nationalists who then fought the Turkish war of independence against said powers resulting in the Treaty of Lausanne in 1923 that established the final status borders of the modern Republic of Turkey, with full independence and no occupation (except for Hatay, if you want to argue that.)

As for why no one attempted to take it after that, well, I don't think restoring Byzantium has been part of anyone's idea of good geo-political strategy for some time. Only the Russians before and during WWI had any inkling of that with their dreams of Tsargrad but that died with the 1917 revolution. Turkey was then neutral in WWII and has been a NATO member since 1952 so I think Istanbul has a pretty good shot of remaining Turkish indefinitely.

The allied powers in WWI intended to do just that. The Treaty of Sèvres would have partitioned what is now Turkey into various zones of foreign influence. Areas with large ethnically Greek populations such as Izmir (Smyrna) and Thrace would be ceded to Greece, and Britian, France and Italy would have large areas of influence. Istanbul itself would still technically be the capital of the Ottoman Empire, but the Ottoman monarchy was basically going to be a puppet of the Western powers, and the city would be administrated by foreign powers as an international port.

You mention that Constantinople is important to Christians historically as the capital of the Byzantine Empire. The Greeks certainly saw it that way; retaking territory historically held by Greeks was a major part of the Megali Idea concept of Greek Nationalism, and using Constantinople as the capital of Greece was a goal of Greek Nationalism in the 19th and early 20th centuries. The Treaty of Sèvres would have helped them move toward that goal, since the Greeks would have been involved in the administration of the city to some degree.

But the Treaty of Sèvres was never ratified; Turkish nationalists fought against the western powers in the Turkish War of Independence and the Greco-Turkish war. These wars defined the modern borders of Turkey, kicked out the British/French/Italians, recaptured the lands ceded to Greece, and ended plans for international control of the city of Constantinople.