Yamaha R6 RJ09 fast rising temperature(~100 °C) under 1-2 km drive

Hello together ! I recently bought a RJ09 Bj 2005. The moped makes a good and clean impression so far. Running performance read off at about 12000 km. For the mileage I would not put my hand in the fire since Englandimport. (Imported 2011) I have the following problem and hope for your help: The moped reaches back within 2-3 min drive or 1-2 km the 100 degrees and that with moderate driving. I had the thermostat in suspicion, since the cooler but rec The temperature of the moped moves between 106 and 120 degrees at the current 10/15 degrees outside temperature. Since the moped was already assumed at my first test drive at home, because the temperature had exceeded >120°C, I assumed that the previous owner did not ventilate the cooling liquid properly and that a lot of water was released. Air was in the system (cooling fluid looked very new). After refilling the cooling system and reasonable ventilation I now have the problem described above and still feel the temperature is quite high. What other suggestions do you have, what do you say about the temperature progression and rapid rise of the temperature, you find the temperature range of always over 100 degrees normal on the road ? Thanks in advance for the help !