I'm not sure what printer or pump you have, but be careful not to push the syringe too hard.
If its like my pump, when the rollers that push/pump on the pipe are in a certain position you wont be able to push, as the rollers will block it. only when it backs of in reverse and it releases the rollers from the pipe can you push fluid through.
Did you try removing the black pipe going in to the pump, running a powerful clean and feeling with your finger for any suction? or popping the end of the pipe in cleaning solution and seeing if anything pulls through in to the drain bottle.
I had concerns mine wasn't working, as it wasn't filling the drain bottle.
It was ether already broken or I broke it pushing to hard testing it, similar to VanderJ test.
Mine had resistance then as the rollers moved (motor) and I pushed I had no resistance, the tube had split, and I had a lovely leak out of the bottom of the pump.
I have replaced my internal pump pipe, but with a pipe not as soft as the original, it turns sometimes but just makes a noise, vibrates and doesn't move for a while. I m currently waiting for a new one.
Does anyone know if bleeding them helps? did you also look if anything was dripping in to your drain bottle while doing a powerful clean?