The Ottoman Empire did not really exist in the 13th century. The Mongols did destroy the Seljuk Empire, which allowed the Uthman/Osmani clan to later dominate the Oghur Turks of Anatolia and then form the Ottomans.
This is an interesting claim because the Mongols and the Huns were both a Turkic people. For example, the Indian Moghal Empire was established by Turks. But many scholars still hold that the Moghals were of Mongol origin. The truth is that the language of the Moghals was Turkic, and that the founders of this empire were proud of being Turk.
Also, the official language of the Mongol Empire was Uigrian, which is a Turkic language. Eighteen Turkish tribes played an important role in the founding of the Mongol Empire. So how you can categorically state that the Huns were not Mongols, based on the Huns being a Turcik people, is puzzling to me.