No legitimacy. They opposed the Ottoman Sultans many times but did not support non dynastic succession as they aren't legitimate. The Giray are almost the same. They don't necesaarily have to be 100% against the Giray, just get deals done. Most other officials would support them as wel.
Assuming the Girayid inheritance of the Ottoman Empire. If they hold the majority of the empire together, might they actually do best in holding the European territories, since that is their own base. Perhaps if the Girayids lose anything, it is the Maghrebi and Arabian extremities of the empire. Would Europe still call them "the Turks" or start calling them "the Tatars". I suspect the West and Central Europeans and Balkan peoples will still call them the Turks, but the Russians may call them the Tatars.
To successfully inherit the empire, the Girayids would have to move their capital to Constantinople. However, with their ancestral connection to Crimea, might their empire be more vigorous in the defense of Crimean and Caucasian territories in the 200 years after 1640, delaying Russian expansion a generation or more?
i'm pretty sure that if the ottomans go extinct after a cold war style moment a civil war between 2 claimants( or even more) will spark, as soon as the main armies are near to fight all the balkans will rebel: Bulgaria, greece/Byzantium, Serbia, Montenegro. The strongest nations would be Bulgaria and Serbia
If on the other hands Girayids fail to take over and hold the realm intact and the Balkan Christians break away - do local nobles or west European royals get picked as their new kings? Is it a distrinctive orthodox sphere or are there vast increases in engagement with western Europe. What about increase in local/regional languages literary output and printed matter? What's the level of anti-Semitism in newly independent states? What amount of the local population already converted to Islam. How many would be massacred, versus flee, versus convert back to Christianity versus be ignored.