It is important to take into context the scale of the war of Greek independence. It was a mostly guerilla effort conducted by feuding rebel groups with very little unity. And although Britain and Russia were involved, it wasn't as though they were committing huge numbers of land troops and occupying major Ottoman cities. Britain's main participation in the war was a single naval battle against the Ottoman fleet, which, though helpful to the Greek cause, is a far cry from seizing the capital city of one of the world's Great Powers. The main reason they joined the war was not due to an idealistic notion of Greek irredentism--rather, as the Ottoman state was in a state of decline, there was a constant threat that the balance of power in Europe would shift. It was in Britain's best interest to participate not because it was the right thing to do for the Greeks, but because it would allow them to prevent Russia from gaining a dangerous amount of influence in the region.

So in the Greek War of Independence, Istanbul just wasn't on the table. It was a relatively small-scale war without the full-throated European support which would be needed to take large amounts of land from the Ottomans. The resulting Greek state was correspondingly tiny, consisting only of the southern part of modern Greece, and the majority of ethnic Greeks remained Ottoman citizens.

This is not directly related to your question, but it might be interesting to talk about the aftermath of WWI, since that was a period where the Turks losing control over Constantinople was a much more realistic proposition. The Greek "Megali Idea" nationalistic concept of retaking western Anatolia and Constantinople was alive and closest to fruition as it had ever been at the end of WWI. Thrace and large amounts of territory in western Anatolia had been ceded to Greece after the war. As for Istanbul, the city was occupied by the UK, France, Italy and Greece for 5 years after the war ended. The Europeans powers in WWI did not intend to let the Turks keep Istanbul--what is now Turkey was going to be partitioned among the western powers (much like the rest of the formerly ottoman-held territories in the middle east would be). This was the plan for partitioning Turkey under the Treaty of Sevres, the treaty which the Ottoman Empire signed at the end of WWI.

Istanbul itself would remain the official seat of the Ottoman government... but the city would be under international control, and the Ottoman government was essentially going to exist as a colonial puppet. Even at the end of WWI it was highly unlikely that Constantinople would be given to the Greeks, as it was a major world capital with important strategic value, and the Greeks were not a major world power--if the allies had their way, it would mostly be administered by the British. But still, the modern Greek state was larger than it had ever been, and Constantinople was in western hands.

So what happened? Why wasn't Turkey partitioned, and the Ottomans maintained as a puppet of the west? Well, the Turkish War of Independence happened. Revolutionaries under Mustafa Kemal Atatürk formed a rival government in Ankara in opposition to the western powers and the Ottoman government in Constantinople. Atatürk was successful against the western powers and the Ankara government abolished the Ottoman sultanate. The Greeks lost the territories they had been awarded in the Treaty of Sevres and a massive population exchange between Greece and Turkey occurred--Greeks in western Anatolia/Constantinople/Thrace were relocated (mostly forcibly) from these areas and made to migrate to the Greek state. Without substantial Greek populations in these areas (which during the Ottoman Empire had always had large Greek populations), any notion of reclaiming these areas for the Greeks became less realistic than ever.