They never dropped the title Khan/Khagan. all Yuan and Qing Emperor still retain their title to the end, even when Mongolia no longer under their power. For that matter, Tang Emperor also claim that title.
I think by 1600 horse nomad would be at disadvantage against gunpowder empire. Ottoman, Safavid, Mughal, Russia, Ming and Qing begin to fight more effectively against nomad with gunpowder. even if you consider Mughal and Qing, nomadic horsemen, they only win with knowledge of gunpowder warfare. Dzungars genocide and defeat of various khanate by Russian show that horse archer no longer most advanced weaponry.