Why are push-propellers so rare, yet they are still around?
There are many disadvantages, they seem to outweigh the advantages.  

Here are two:


A pusher prop is working in a disturbed airflow, causing increased vibration and noise
If the propeller is fitted behind a wing, each propeller blade is passing through the separated boundary flow twice each rotation. These cycles create additional noise and lower the efficiency of the propeller. The vibration makes the propeller blades more susceptible to metal fatigue. 
Propeller clearance at takeoff
Due to the pitch up at take-off, the propeller gets close to the ground. Therefore the diameter needs to be reduced (loss of efficiency) or the landing gear struts need to be made longer (added weight). Since the propeller is behind the landing gear,  it is susceptible to debris kicked up from the gear, increasing the need for added blade protection (increased weight, loss of efficiency)


Wikipedia has a list of additional disadvantages. 



Edit: 

Your statement that 


  Jet engines are by their very nature push-engines


is not entirely true.

In turbofan engines, most of the thrust is generated by the fan and compressor stages. Even in a pure jet engine, a lot of thrust is generated by the compressor. Therefore the shaft of a jet engine is tension loaded, just like a propeller shaft in puller configuration. 


Source: Rolls Royce - The Jet Engine