Yes, Arabic script was a spectacularly bad fit for Turkish. But a more proximate reason, surely, was that mass literacy presupposes printing—and the Ottomans did not allow printing in a Muslim language. (They didn’t allow it in Christian Greek either, but at least Venetian printers were able to capitalise on that.)
Global spread of the printing press
Due to religious qualms, Sultan Bayezid II and successors prohibited printing in Arabic script in the Ottoman empire from 1483 on penalty of death, but printing in other scripts was done by Jews as well as the Greek and Armenian communities (1515 Saloniki, 1554 Bursa (Adrianople), 1552 Belgrade, 1658 Smyrna). In 1727, Sultan Achmed III gave his permission for the establishment of the first legal print house for printing secular works in Arabic script (religious publications still remained forbidden), but printing activities did not really take off until the 19th century.
1727: First press for printing in Arabic established in the Ottoman Empire, against opposition from the calligraphers and parts of the Ulama. It operated until 1742, producing altogether seventeen works, all of which were concerned with non-religious, utilitarian matters. Ibrahim Muteferrika.
1779: Abortive attempt to revive printing in the Ottoman lands, by James Mario Matra (Briton).
[Btw, Wikipedia? What printing in Greek communities? Judaeo-Greek doesn’t really count as the “Greek community” in this context. Citation needed.]
On the late adoption of the printing press in the Ottoman Empire
It is clear from the historical evidence that the professional manuscript scribes were violently against the printing press because they did not want to lose their jobs.
It is not so clear that the clergy of that era were against the printing press even though historians show evidence that it was the case. In the current intellectual climate of religious fervor in Turkey the prevailing opinion is that the clergy of the 16th, 17th, 18th centuries were not against the printing press. They say that the printing press was rejected because of the strong opposition of the calligrapher/scribe class. This begs the question: if the clergy were not against the printing press why Muteferrika or others were not allowed to print religious books?
Myths and reality about the printing press in the Ottoman Empire (which looks like a weak apologist account to me—they didn’t allow it because it wasn’t as pretty as manuscript? And only the scholars needed to read anyway? Really?)
Printing banned by Islam? (Christian polemicist, but consistently with scrupulous scholarship)