Seeing as Pu Yi was the last emperor of the Q'ing dynasty, and the Q'ing was directly preceded by the Ming and Yuan dynasties, was there a real unbroken line of succession of emperors between Kublai Khan and Pu Yi? I can't seem to find any period between the start of the Yuan and 1912 when there wasn't an emperor on the throne, and it would make the end of the reign of Chinese emperors so much more significant if Pu Yi really was the last emperor in a line of emperors that goes back more than 700 years. Or were there moments where China was without emperor between 1279 and 1912?
Im wondering why is the Yuan is the demarcation. The transition between the Southern Song and the Yuan is about as seamless as the transition between the Yuan and the Ming. The Northern and Southern Song and the same dynasty, just differentiated when the moved capitals.
To answer your question though. There was always pretty much an Emperor at that time. A few didn't rule but they existed. Tongzhi was technically an emperor but never ruled. A few of the last Ming emperors never really ruled either like Longqing who let ministers handle everything for them. So the answer is no, your correct there was always a sitting emperor during that span, except for a few short periods that only lasted months.