Forgot to add that at reduced weight aircraft which require a functional autofeather (or equivelent) may depart in compliance with the CAO;
(Very simply) To climb Thrust and Lift must balance or exceed Weight and Drag. Assuming power is limited then if drag dramatically increases (windmilling prop) then by reducing weight you can get the required performance. But the weight penalty is usually quite large.
Engine failure with the failure of an autofeather system at high weight soon after V1 and some large turbo-props may even struggle to get airborne at all let alone climb after lift off. Quick action by the crew to manually feather the prop is required.
Most twin engined aircraft could reduce weight to a level where able to climb in various high drag situations. But are these statements still true at normal operating weights? Most likely no, and configuration becomes critical.