My friend
At a constant rate of turn at a constant altitude, there is a constant acceleration towards the the axis of the turn, and a constant acceleration upwards to match gravity (whose vector is hardly likely to change over this scale). There is no energy change and no requirement for power change.
A constant turn
is acceleration. You are changing your velocity vector. I have flown fighters for many years and I can tell you that a 4G turn requires a lot more power than just cruising along at 40 000 feet
Anyway, regarding the AAIB report:
b. Causes
The following causes were identified:
(i)The handling pilot's failure to recognize the rapidly changing relationship between airspeed and groundspeed which is a fundamental problem associated with turning downwind in significant wind speeds
(ii)The Commander, who was the handling pilot at the time, shortly after takeoff inadvertantly allowed the airspeed and then the height to decrease whilst turning away from a strong gusting wind
(iii)Despite the application of maximum power, the helicopter was incapable of arresting its established descent within the height available. Incipient Vortex Ring state and down draughts may have contributed to this problem, as may the height of the wave crests.
Can we now agree that the turn downwind was a factor?