Why did they stop where they did considering there were many more cities and Greek people throughout Anatolia?
Constantinople was a major goal of Greek nationalism in the late 19th and early 20th centuries. Greek ambition for territory in Anatolia, whether Istanbul/Constantinople or Izmir/Smyrna, pretty much ended after the Greco-Turkish War, one of several wars in the region in th wake of WWI. During this war, Greece occupied and administered Izmir for several years and had their armies reach the Istanbul suburbs. The tides turned decisively, however: In Turkey, the war is seen as the most important part of the Kurtuluş Savaşı, the Turkish War for Independence (literally the War of National Salvation), whereas in Greece it’s know as the “Asia Minor Disaster” (Asia Minor is the Greek term for Anatolia).
For more on both this and the larger context, see my old post:
When the Greeks initially became independent in 1829 they were only capable of securing a part of modern Greece, they would not have been capable of invading Constantinople or Western Anatolia. If they were capable of doing so maybe they would have.
In the aftermath of WWI it was the original intention of the Allies to give Greece the parts of Western Anatolia that were populated by Greek people and make Istanbul/Constantinople an international zone. The allies also wanted to give other parts of modern day Turkey to other countries. The Turks rallied to regain their Anatolian territories and pushed the Greeks out of Western Anatolia. They ultimately negotiated a new treaty in 1923 which left the Greeks with no territory in Anatolia and resulted in population transfers that left the Greek and Turkish areas adjacent to the Aegean Sea fairly ethnically homogeneous.
They tried. Also, Greece does see itself as continuing the Byzantine tradition but this is not without controversy. There is a 600 year gap between Greece, with a German king and supported by European Powers, and the Byzantine Empire— the “east”, oft-enemy of Europe, and inheritor of classical Greece. That many Byzantines served the Ottomans loyally and the Ottoman Sultan was declared the Byzantine Emperor and also claimed Byzantine royal blood should be noted. Mehmet II claimed that he merely toppled a pretender Palailogoi Emperor to protect the trampled upon Roman people and to assert his Komnenoi heritage (his grandmother). Moreover, many Greeks died in trying to capture parts of Turkey in 1924. This was called the “great catastrophe” in Greek because of the total failure. Perhaps every family in Greece lost a father, son, and/or brother in Anatolia to Mustafa Kemal’s army.
There are very serious historians who believe that the Ottoman Empire is really just an updated Byzantine Empire or Roman Empire v3.0. So in this view, Greece has no special legitimate claim to the Byzantine legacy. They don’t even respect the Patriarchate’s control over most of the country.
Plus the whole German king thing is a big deal. Greece was a European project from the start, not incomparable to the pretender client states of the Latin crusaders.