The primer only works if there is pressure in fuel system...either by bulb pumped up or by fuel pump when engine is cranked. To release pressure into carb/manifold fittings the key has to be pressed.
Cylinder can be damaged and still have good compression. A compression test only shows that a cylinder has enough pressure to support combustion but not the condition of engine. Rebuilt many a motor with good compression but bad cylinders
It is not a pump ..its a valve that opens with pushing in on keyswitch. The pressure in the fuel system and the cycles of the fuel pump is whats makes it "squirt"