This is actually quite common. It's a phenomenon called "Middle Man Minorities". Look to, for example, the Jews in Europe or the Overseas Chinese in Southeast Asia or South Asians in East Africa (Idi Amin expelled them all from Uganda, but there are still large communities in Kenya and Tanzania). I strongly recommend taking a look at this post (explaining the Jews, mainly, but also delving into things like the Koreans being targeted the LA Riots because they were a middleman minority in South Central L.A.) because I deal with it more generally.

As for how the Armenians and Greeks (I don't know about the Copts--I've never heard that before) ended up as middleman minorities is harder to explain. One part has to do with the Ottoman education system. As Europeans become more involved with the eEmpire, minorities ended up minority schools built on European lines--often funded by European co-religionists--so they had much earlier access to modern European education rather than traditional madrassa education. Ottoman civil service schools were only set up in the late 19th century, and they mainly trained people for state service rather than people involved in private business ventures.

But that's the other thing--the Muslims in the Empire were much more likely to invest in land than businesses. Most (but by no means all) of the big landlords in the Empire were Muslim. Further, by the early 19th century, after the crushing of the Janissaries and the beginning of the Tanzimat (reorganization) era, promising young Muslims often ended up working for the state, especially for the military. The vast, vast majority of both the Young Turks (who ruled the Ottoman Empire from 1908-1918) and the Kemalists (who ruled 1923-1950) were trained in the modern Ottoman schools I mentioned above, and had early careers in the state, particularly the military.

In addition to things like intra-ethnic trust fostering trade ties over large distances and easier access to credit (something that was always difficult to get in Muslim communities), the minorities often had certain legal advantages when it came to doing business. I'd have to look up the details, but international treaties ended up giving minorities some advantages in international trade, starting with the Greeks in the 18th century but expanding quickly to other groups (the traders, though technically Ottoman minorities, were granted sorts of protection by foreign powers as part of the capitulations--capitulations to genuine foreigners began earlier). So just as Muslims had legal advantages in land ownership and service to the state compared to non-Muslims, minorities could get legal advantages in trade compared to Muslims.

In short, the minorities often invested in skills and later capitalist businesses, while the Muslims often invested in land and later careers within the state. While for most of the Ottoman Empire, the latter was the more effective strategy, as the Empire lost power as capitalism and the world economy became increasingly important, the minorities were much better positioned to take advantage of the changing system. But the existence of middlemen minorities is also incredibly common around the world. Over large swaths of the Indian Ocean, for example, Muslims and Arabs were at various times themselves middlemen minorities. It's just one of the ways that long distance trade networks were possible before mass communications, international banking, and globalization more generally.

I'll give a more detailed response later tonight, but there were a number of reasons. First was Islamic law's requirements on inheritance. Islamic law doesn't allow bulk inheritance: instead, property is distributed among relatives. While Muslims did find ways around this (most notably by putting the money in Waqfs, charitable trusts), overall it had the effect of making it much more difficult for Muslims to accumulate the wealth needed for large scale commercial ventures. Continuing on that trend, interest banking is forbidden in Islam, further limiting the capital available to Muslim businessmen. While the Ottoman Empire didn't follow Sharia strictly, Ottoman and Muslim law frequently were similar, and even though the law may have permitted some practices, customs or local laws may have differed.

Secondly, Christians enjoyed a historical advantage. Muslims were the (relative) newcomers to the great coastal trading cities of the Middle East, while Christian groups had long-established relationships and expertise within and without of their communities. The "default" was these Christan groups having an advantage, and it was more difficult for Muslims to break into the business.

Third, and probably most importantly, the Christian minorities had a great advantage in trading with the non-ottoman world. In the early days of the empire Christian minorities, especially Greeks, were the middlemen between the ottomans and Europeans (especially venetians). Greek was a lingua franca thanks to Mediterranean trade networks, and Greek interpreters and assistants were widely used on both sides. This greater contact with the west gave Greek (and other minority groups, like Jews) a leg up in trade, as they established relationships. Later in the Empire's history this continued, but in a different way. Christians who worked for various embassies and consulates were able to get a form of diplomatic immunity and protection from powerful European countries, especially in the commercial sphere. Ottoman-European trade was governed by a sort of unequal treaty system, called capitulations. Originally they were established to ottoman benefit by easing trade, but later on they were revised and were a negative factor for the ottoman economy. But Christians working for consulates got the same commercial privileges as their European benefactors, aka lower taxes and tariffs, giving Christan minorities another leg up.

I'll expand and elaborate my explanation later tonight

The question assumes that the Ottoman Muslims were the majority. It wasn't till the 19th century that the Ottoman Muslims overtook the Ottoman Christians and otherwise in population, mainly due to the wars which caused the loss of most of Ottoman Europe, holding its large Christian populations.

ohh, Lewis, one of few western scholars denying the Armenian genocide, sitting on all kinds of turkish boards and receiving grants from the republic. I don't take any of his research for serious, a sold out soul.