1- Ottomans do not lap with the medieval period much. Your logic of representing them in two ages isnt logical. If anything I would say Gokturks+Seljuks. Because Seljuks themselves were high medieval period. Or simply a synthetic Tengrist Turkic culture(Bulgars,Avars,Gokturks,Cumans) for first ages and Seljuks-Turkmen beyligs for the last ages. Do not forget conversion of Turks to islam happened very late in medieval(11th to 14th centuries gradually)Ottomans were an early modern period empire. If we take medieval period as 476-1453.
They were founded in 1300 but until 1396 call it 1400 they werent anything significant in Turkic world. What is more, until 1444 they werent any different than other Turkmen states, in 1444 battle of Varna they first owned their victory to janissaries. Still janissaries were just archers at that period, they didnt adapt muskets much.
If Ottomans are going to be used, wait for a game that is based on renaissence era.
2- They are very distinct from other Turkic empires. They do not rely on cavalry that much after their foundation period(their foundation period culture isnt much different from other Turkmen states anyway). They are unique on their own and that uniqueness comes after medieval period. They were a gunpowder empire.
Western world focuses on it too much because Ottomans were partially in Europe but compared to the the medieval period of Ottomans Timurid was for example far more significant.
3- They are overpresented/overportrayed. I am personally sick of seeing “janissary” as Turkish unique unit. They werent even Turkish by origin. Thats not the case however, when all medieval Turkish empires, conquerors etc are considered they are just so away from Turkic warfare culture.
One would expect a horse archer or light cav unit for instance.
One might say there will be more than one unique units however I think it is too specific/distinct/far from the core even for being one of the Turk unique units.