Dave,
Your assessment of the physics would be correct if the lift vector was a constant. It isn't.
Generally, the hover is the most power hungry mode of flight. The wonderful world of RW aerodynamics enjoys dramatic increases in rotor thrust with forward speed, almost identical with that of aeroplanes.
The power required vs airspeed diagram looks very much like the diagram you are more familiar with. It matters not that the causes are different, the outcomes are largely the same. One accelerates for take-off to maximise the reduction in power required - things really start to happen around 10-15kt, max angle is generally around 35-45kt, max rate is around 55-74kt and above that you really start to pay for the privilege of going (relatively) fast.
While on the subject of power required charts, the multi-engine issue is exactly the same as for your twin aeroplane: lose one engine (twin, not EH101) and you lose between 75-95% of your performance.