Ottoman History major here. I, myself played EU4 for a long time as I did with many paradox games. I believe that they are working in cohort with actual historians however they do so thoroughly as they introduce new DLCs and Harem is in play for couple years now. I want point a specific wording on the "Harem" and would like to answer your question as much as possible after that.
What I want to point out is that the specific wording in EU4 official wiki page which is " The harem helps ensure that there will never be a lack of heirs for the empire." In this case I believe that in a development sense, the Harem does not let you choose an heir but it prevents lacking of an heir. Which is then supported by the flavor text that implies that there are many capable heirs therefore the choice between them.
Now, if we look at the actual historical facts, Ottoman Sultans do favor one children over another as they use Lalas (personal instructor of an heir) and Sanjaks (Regions) to slightly boost one heir over another. For instance a well connected Lala and a Sanjak that is closer to the capital indicates that current Sultan is favoring this particular child over another with a farther Sanjak and lesser known Lala. But to answer your assumption/question that Sultans have more control than Europe in choosing the heir especially in 15th century, the answer must be: No they don't at least in the first half of the 15th century and yes they do after 1444. After the 1402 defeat of Ottomans at the hands of the Timurids, there is a period of interregnum that is free-for all between different heirs for instance until 1413. Even after this interregnum period there is another "pretender" Sultan to the throne of Mehmed I as later fratricide was the only option for him (Mustafa Çelebi). However as we move to the middle of the 15th century, especially 1444, we see one thing change, that is to say the coronation of Mehmed II (the conqueror) and his subsequent purge of the other power centers from the empire. With this purge and stricter controls over heirs, the will of the Sultan becomes nearly absolute (at least until 17th century where Harem's power and certain families' power becomes prominent again in politics). At this time favorite heir of the palace has so much power that he can either negotiate with his brothers to pursue a life full of Islam, wealth and no power or the alternative which is fratricide by choking (since the bloodline of the Sultan is holy, it should not be spilled). Although the rule of primogeniture is there, much like Christian polities, however it can be ignored in certain situations.
As my closing remarks I want to point out that, although in this specific time period of Ottoman history (late 15th to 17th centuries) it is comparatively easier for a ruler to select an heir, it is not so in general. All polities from around the globe has such instances where selection of heir is simple or as complex as it can be and most of the time it is not up to the ruler to select. It is a combination of heir's charisma, his connections, power centers that are interested in the heir, ruler's power and image, luck (bubonic plague, etc.), tradition and many other factors.

edit: grammar and spelling

The path between the name of the mechanic and the actual historical mechanism is a bit torturous, though clear enough once you read it. But executing it as selecting an heir is...not really how it went. More can always be said on the matter if anyone would like to go into Ottoman succession law; for the meantime, OP, u/Chamboz covers the Law of Fratricide.