Well, for the most part, the Turkic roots of the dynasty were clear enough. This is clear in Mehmed Neşri Efendi, a 15th-16th century chronicle who quotes Murad I (from the 14th century) as replying to the Serbs: 'God willing, I shall show them Turkish manliness'. Ottoman genealogies also start the history of the dynasty from Oguz Khan, a mythical figure that was the ancient leader of all Turkic peoples.

That said, you are right that sometimes 'Turk' had a pejorative use too. This was intended to capture those Turks that lived in villages of Anatolia, or worse, as nomads. A common derogatory term for these Turks was 'etrak-ı bi-idrak', meaning 'the undiscerning Turks'. Similarly unkind terms were not spared from many other ethnicities forming the Empire; but the Turkish case clearly shows that while there is an awareness of Turkishness and the Turkic roots of that identity, it is not quite a full-blown nationalism yet.

The Muslimness was quite important for Ottomans too. Contrary to what some people nowadays seem to think, they were quite clearly no angels in this regard as they were known to drink and gamble from time to time. But nonetheless being a Muslim state was very important. Their drawing from Persian culture is in part due to this, as Persians are Muslim too. Many Ottoman literary traditions and institutions also have their roots in Caliphates.

There is also another aspect that got lost in time. That's what is sometimes called 'the Alexander romance'. It was in fact quite prevalent for Ottoman rulers (and, I assume, for some other Islamic rulers before them) to read about Alexander the Great's life and it was also common to draw parallels between them and Alexander. Alexander the Great himself became a sort of legendary figure with time, of course, whose life was recounted in numerous ways in many parts of the World. I will not dwell much on this one since you are probably more interested in what kind of an identity Ottomans attributed to themselves, but I'll refer you to a lovely speech by Dimitris Kastritsis of St Andrews on this issue.

Ottoman Empire existed for around 600 years. Understandably, the understandings of Ottomanness changed with time too. To my knowledge, the most vibrant period of discussion about 'who are we?' occurred from the mid-19th century onwards, when an Ottoman identity (called Ottomanism) which would in paper exceed all ethnic and religious loyalties was seemingly becoming popular. But this did not become the American identity for the Ottoman Empire; that is, Korean Americans and African Americans are all Americans and tend to feel loyal to the US, but that did not work out for the Ottomans. Hence the rise of nationalisms, and also of Islamism. The idea of the Ottoman Empire as a Muslim entity was always strong, as before, but now it was seen as the formula that could keep Albanians, Turks, Kurds, Circassians, Bosnians (etc...) together. This period is also interesting in other aspects. Şemseddin Sami Efendi (or Sami Frashëri in Albanian), for instance, shows brilliantly the spirit of the time. He came from a proud and noble Albanian family. He even wrote the first Albanian play and had his share in inspiring Albanian nationalism. It is the very same guy who opposed in the late 19th century attempts to define the Ottoman Turkish as simply 'Ottoman' and argued it was just Turkish. He then wrote an excellent Ottoman Turkish dictionary that historians still use a lot, and called it Kamus-ı Türki (the Dictionary of Turkish). That said, I suppose this is not very surprising in empires of the 19th century; I would not be surprised at all if people existed in Austria-Hungary who felt, say, Czech, and at the same time felt some degree of loyalty to the Habsburg Empire.

Speaking of the Habsburgs, Ottoman sultans indeed used the title of kayser-i Rûm (Caesar of Rome or Roman lands). The use of this title apparently began only with the rule of Mehmed II (the Conqueror) who took Constantinople. Especially under Suleyman I (the Magnificent), that is in the mid to late 16th century, the use of this title grew in importance. You have Ottomans denying the right to use the title of caesar/Kaiser to the Habsburgs and Germans at this time. This most likely signifies a claim to becoming the 'world empire', as it were. Ottomans insisted on this for quite a long time as this was a tool of supremacy over the German-speaking entities with whom the Ottomans were in conflict at the time. Beyond this, though, did the Ottomans consider themselves as Romans? Not in origin surely, and not much with regard to the Roman heritage either.

This comment became longer and much more rambling than expected, so here's the TL;DR. Ottomans considered themselves as a Muslim entity, for most of the time they knew their rulers were of Turkic origin and these rulers themselves seemed to care about it occasionally, but the Turkish identity was surely not as pronounced as it was in the 20th century. Persian influences were surely there in Ottoman culture and art, but I am not aware of any cases of Ottomans actually believing themselves to be of Persian origin. The title of 'The Caesar of Rome) was a real thing, but probably does not mean that Ottomans considered themselves to be Romans in origin or even in heritage (if not for some exceptions).

Sources and Recommendations:

Dimitris Kastritsis - The Alexander Romance and the Birth of the Ottoman Empire, 26th January 2016 at the University of St Andrews Institute of Intellectual History. https://intellectualhistory.libsyn.com/website/dimitris-kastritsis-the-alexander-romance-and-the-birth-of-the-ottoman-empire

Işıl Demirkent, 'Kayser', TDV İslam Ansiklopedisi, (2002). https://islamansiklopedisi.org.tr/kayser (Accessed 29th May 2020)