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After the Ottoman Empire collapsed, why did Turkey (i.e. its successor state) retain control of the Empire's Balkan holdings? Considering the historical origins of the Empire Turkey should have gotten its holdings in Anatolia and Asia Minor, while Istanbul and the Balkans should have been given to Greece (after all, Constantinople had been the seat of the Eastern Roman Empire for about a millennium).

Looking at the map now, Turkey literally holds what used to be the territories of the "Byzantines" (I don't like that word myself).

The borders of Turkey were established first by the Treaty of Sevres and then by the Treaty of Lausanne. Turkey lost as a party of WWI, and the negotiations followed.
Turkey did not accept the first of the mentioned treaties, and a war followed. Essentially the Turks won this war. The main loosing party was Greece, but the Western Allies did not want to or could not make sufficient effort to win this war. The compromise was achieved and fixed by the treaty of Lausanne.

The great power which wanted Constantinople and surrounding area was Russia which posited itself as a successor of the Byzantine empire.
This desire of Russia was actually one of the principal causes of WWI. At some point during the war, the Allies were inclined to give Constantinople to Russia. But Russia was out of the war, and not a "great power" anymore, at the time when the war ended, its own empire collapsed, and there was no strong enough power willing to fight for Constantinople, so the allies agreed to leave it with Turkey. The Greeks were willing to fight but they were defeated.
See Magali Idea about original plans of partition of defeated Turkey, after Russia was out of war.

So the short answer on your question: "The Turks fought hard for this territory and won". In fact the very existence of Turkey was in question. How could they win against all odds, is a separate question. But the main reason is that the Western allies had no will to fight in the 1920s, while Russia had more urgent things to worry about. And the Greeks were not strong enough.

Remark. In general, the borders are not established by "what is fair", or what is "reasonable", or "what belongs to whom, historically", or even what is the majority of populations, etc.
Borders are the results of wars. At least the Old World borders.

Let's look at the map in 1913:

(click for large)

Detail view 1913:

^{Omniatlas: Europe 1913: Aftermath of the Balkan Wars}

^{"Greece according to the Treaty of Sèvres", in Wikipedia: Partition of the Ottoman Empire}

These tiny remnants of Rumelia on both maps are not overly meaningfully described as "Empire's Balkan holdings".

The Empire's Balkan holdings, or Rumelia, looked as recently as 1861 more like:

Or even in 1912:

So, after the Great War, Turkey only emerged from the ruins of the Ottoman Empire and just held the line at its western border? Compared to its former "Balkan holdings", this is a really tiny strip of land in the historic Marmara region.

During the Turkish War of Independence and especially the Greco-Turkish War (1919–1922) the chaotically mixed settlement patterns of Greeks, Turks, and 'others' made it not overly convenient to base any considerations for future borders on 'historic precedent', 'rights' or even 'majority populations'.

Contemplating something like "Istanbul and the Balkans should have been given to Greece" – for whatever historical reasons, going back to late antiquity even – would mean that any armchair diplomat proposing such would ignore the past 800 years of historical development and the then present situation on the ground. That is of course great for nationalist expansion and a terrible idea for peace.

To rephrase the initial question:
Q Was there a reason why Turkey retained the rest of the Balkan territories of the Ottoman Empire, instead of adding East-Thrace and Istanbul to Greece or another of the Balkan states?

Yes. The Paris Peace Conference had terrible ideas for the future borders of a post-Ottoman/Turkish state. That caused a war in which the participants wanted to decide the argument with force. Turkey emerged from that war less defeated compared to other prying eyes and was able to largely retain the borders in its West as agreed upon in the concluding Treaty of Lausanne.

The Greek front collapsed with the Turkish counter-attack in August 1922, and the war effectively ended with the recapture of Smyrna by the Turkish forces and the Great fire of Smyrna.

As a result, the Greek government accepted the demands of the Turkish national movement and returned to its pre-war borders, thus leaving East Thrace and Western Anatolia to Turkey.

The Allies abandoned the Treaty of Sèvres to negotiate a new treaty at Lausanne with the Turkish National Movement. The Treaty of Lausanne recognized the independence of the Republic of Turkey and its sovereignty over Asia Minor, Istanbul, and Eastern Thrace. Greek and Turkish governments agreed to engage in a population exchange.
_{WP: Greco-Turkish War (1919–1922)}