Sigh. Okay the only thing the engine is spinning is a hydraulic pump. The pump is positive displacement, meaning that what goes in must come out or it explodes or stalls.
Ask yourself, if the engine is running faster or slower, if there a difference in flow?
Answer: yes, there is a difference in flow.
Ask yourself, is the pump able to go to relief pressure at idle?
Answer: yes. It is possible to nab something with the loader at idle and go to relief.
Ask yourself, why then is more lifting power available at high engine speed?
Answer: the volume of flow. The speed of flow allows more work to be done even as the relief is varying its opening and blowing off excess pressure. This is because pressure on fluid is the same everywhere when the fluid isn't moving but this isn't true when the fluid is moving. At low volume, a given ram will not move under a given load because the relief opens before enough volume of fluid can develop to overcome the load. At higher volume, enough fluid is moving to fill the ram to keep it moving even as excessive pressure is flowing past the relief valve. Thus more work is work is done with higher volume and higher engine power allows the work to be done faster on a given ram size than a lower engine speed or lower engine power. For this reason the 45-60hp NX series have the same spec loader lift spec ratings.
I tried to find an animation to show how this works as it isn't super intuitive. What I found is a closed center system animation. Fine, just presume, a valve that flows pump output back to tank when not working the valve rather than through the relief. what I liked about this animation is that it attempts to show pressure under different loads while the ram is moving and the speed at which work can be done. The animation doesn't get into fluid dynamics and Bernoulli's Equation, which is what I really wanted to try and show.
The basic take-away is that higher flowing fluid has lower pressure than slower moving pressure and so more work can be done before going to relief.