No.
Firstly hot water rises.
It is true that the water wouldn't travel up if there were an air pocket in the upper part of the radiator. In that case the water would exit the lower exit valve without passing through the upper part of the radiator. The top part of the radiator would feel cold. The radiator would not adequately heat the room.
If the radiator is full of water then you should remember that it acts as a heat exchanger, hot water entering the radiator is cooled and nearby air and the room are heated by conduction, convection and radiation. Cooler water is denser and sinks, incoming hotter water is less dense and rises.
Secondly remember that when first fitted, radiators are full of air and, with the inlet and outlet valves open, you open a bleed valve at the top of the radiator to let the air out so that the radiator can fill. You close the bleed valve when water starts to seep out without air bubbles in it.
This process requires the water in the heating circuit be at higher pressure than local atmospheric pressure.
In older systems this pressure was achieved by having a header tank in the roof space about two meters above the highest radiator.
In newer systems (or also older systems in some parts of the world) The pressure is achieved by pressurising a closed loop heating system from mains-water into a small pressure vessel.
Also the water pump works by pushing water around the system and this raises the pressure.
TL;DR: Your heating system is pressurised and hot water rises.