I highly recommend Bosnia: A Short History by Malcolm. He does a good job of introducing the subject of deviširme and explains a lot of context in how it effected Bosnia and the Ottoman Empire.

Long story short is the "abductions" were initially met with fear and sorrow at having mostly sons taken from their parents and taken to modern day Turkey. The kids were educated and trained in various political roles and (usually) eventually allowed to return back to their home town. Once the families realized their kids would be returned as wealthy, educated professionals the parents began trying to offer their kids for this selection.

Bonus for this book is it goes a long way for providing context for how WW1 started and valuable (and unbiased) background for the history of Bosnia which was systematically destroyed by those who might benefit by fabricating a story that Bosnia was never a country and Bosnians were never an homogenous people of their own culture.

The post body doesn't mention devshirme, so I'm going to pretend it's not in the title and answer more generically about the prominence of non-Muslims (excluding converts) in state service during the late Ottoman Empire.

It's clear that non-Muslims were not prominent in the Ottoman military with the exception of Greek sailors serving in the navy. Non-Muslims were not wholly excluded, however, and in the twentieth century, you start to see them serving in the military proper, if only in small numbers. During the First World War, non-Muslims were often forced to serve in non-combat roles and notoriously in labour battalions under harsh conditions. Nevertheless, Ottoman-era military graveyards still show the occasional Greek, Armenian or Jewish soldier. I also know of cases where non-Muslims were to be found serving a military doctors; unfortunately I do not know if anyone has studied what percentage of the total they constituted.

Outside the military, it was easier to see the non-Muslim presence. Many of the 19th century Ottoman palaces or other significant buildings were built by Armenian architects: Dolmabahçe Palace and Ortaköy Mosque are two notable examples that many people will be familiar with. The bureaucracy was additionally populated with many non-Muslims. The translation bureau was traditionally dominated by Armenians and Greeks. And the diplomatic corps was also significantly non-Muslim, with envoys to many European capitals being Christian. One of Ottoman ambassadors in Washington was also Greek. A book that painstakingly lists Armenians working as civil servants in the Tanzimat and Hamidian periods is Krikorian's Armenians in the Service of the Ottoman Empire. I don't know if there is a similar work concerning Greeks, but a quick trip through an Ottoman sâlnâme (many of which you can find online) shows many non-Muslim names in all sorts of positions.

It's also worth mentioning the breakdown of the Ottoman parliaments since there was a significant non-Muslim representation. Over 20% of deputies were non-Muslim in the 1908 parliament, for example.

Despite all this, there was clearly an "upper ceiling" to how high non-Muslims could rise; no Christian ever became a provincial governor (although there were vice-governors) or grand vezir (although there were ministers). The good news is that there is a rich scholarship on the experience - positive and negative - of Ottoman non-Muslims (especially from the "big three", Greeks, Armenians and Jews), especially for the 19th and 20th centuries.