Possible overheating

[Faber32]

Good morning guys  Yesterday I replaced coolant on my People Gti 125. I made a mistake. When filling new coolant, I started the engine for some seconds (classic procedure to bleed air), revving up lightly, topping up while radiator level was going down and after shutting off the engine I opened the bleed screw near thermostat to bleed air from the circuit. After 6-7 times of this start&stop, air was still coming out from the bleed screw. It seemed strange to me so much air was still trapped, so I started the engine and let it idle for minutes to see what happened then. I was waiting for the fan to start, but it didn't. How can I know if the cooling fan is broken without a diagnosis scan tool?
Also, after 7-8 minutes the temperature went from the safe level 2, to level 3. I immediately turned off the engine.  Then it happened a second time after, evidently there were still some air bubbles inside. Or it was simply overheating because it was idling still and fan not working. At some point also the CEL appeared but I turned off immediately. Do you think I could have damaged the engine? Or the new coolant itself?
Thank you

[Ruffus]

Assuming you filled via radiator cap.
From your scoots waterpump departing two hoses. A thicker one and a thinner one.
Take the thinner one partially off and bleed the system with this one. To drain fluid, no need to open anything at the waterpump!
Let it rest open (radiator plug) overnight.
 (refill  and bleeding see pics)

Checking fan: two wires lead to the fan
-  unplug, apply external 12V to this terminal

One more thing: thermostate could be faulty or stuck.

[Faber32]

Thank you Ruffus!
On my People, the water pump has a drain screw. But not all the coolant comes out this way. You have to lift the front wheel very much to empty totally the circuit. My only fear was of overheating the head gasket, but probably it requires more to make damages