I think you missed the point of the author. Descent from Genghis Khan was prestigious because Genghis Khan conquered half of Asia in his lifetime. Descent from Osman was not, because Osman conquered and ruled less than half of modern Bursa province at the time of his death. The actually prestigious Osmanlis were the great conquerors Mehmed II, Selim I, and Suleiman I, precisely because of their conquests.

Timur had no prestigious descent either. He spent his life and reign claiming legitimacy as a servant of God and the Chagatai khans Suurgatmish and later Suurgatmish's son, Mahmud, who were actual descendants of Genghis Khan through his second son, Chagatai. Timur's son Miran, and later his grandson, Ulug Beg, married daughters of Suurgatmish and Mahmud. Because of this, the Timurids and later the Mughals actually called themselves the Gurkani, which translates more or less as "the Sons-in-Law". For themselves, their prestige was not in their patrilineal descent from Timur, as their English name suggests, but in their matrilineal descent from Genghis Khan himself.