We will see friend: The water of the expansion bottle must rise and lower depending on the pressure generated in the cooling circuit due to the effect of the temperature of the refrigerant element, that is: if the temperature increases, there is more pressure and consequently the water expands towards the bottle. If the temperature drops, pressure and water decreases returns to the circuit, so the level of the expansion bottle will not always have the same level.
I am struck by the fact that you comment when you say that "... if you start the car without the bottle plug within a few minutes the water flows towards it ..."
well, this could be normal; Normal because in theory the water has already heated enough to overcome the resistance offered by a valve that is closed by the action of a spring and that is usually found in the radiator filling cap.
I say that this behavior could be normal if the premises commented above are met, but they would not be so much if they are not met, then then it would be necessary to suspect the effectiveness of the radiator or valve plug valve that also carries the expansion bottle to achieve the “thermosiphon” effect. (But this is another story)
apart from this whole “roll”, I must reiterate the above commented by the fans: first to simple, although… we must not forget something very important and that although it is necessary that the radiator must be clean internally, it is no less that it must be so much or more on the outside.
I want to say that with a water jet with a little pressure, clean the radiator on the outside, remove flies, and bugs that are deposited and obstruct the circulation of the air that has to cool the pipes that contain the refrigerant liquid.
Finally, do not forget to take a look at the alternator's strap that in turn moves the water pump, verifies the tensioning status of the belt and checks the accommodation of the pulleys checking that they are clean of oil so that the belt does not skate on them.
This in a gross way, because the truth is that the issue of heating "gives a lot."
Receive a cordial greeting and you will tell us something.
Zulu