As usual, there have been a lot of generalisations made about the flight and pilot, most of which do not actually help identify exactly why Pete Barnes hit the crane. And as with all accidents, the exact, ultimate cause is important and often not focussed on enough, I feel.
Ironically I think he hit the crane because (at this point) he was trying to comply with the rules - or at least to the greatest extent he could. The pilot had made a fairly tight right hand turn over Chelsea Bridge, hand flown, at a fairly steady altitude, so was almost certainly visual here. He then descended quickly to 570ft as he was proceeding east, again almost certainly to maintain VMC with a lowering cloud base as he went east. At this point, given clearance to turn west for Battersea Heliport, I'd guess that he knew turning right over the river at 500ft, whilst perfectly safe, would put him blatantly and obviously in breach of Rule 5, so he chose to climb as high as he could whilst being able to see the ground, which he surely could, given the images showing the silhouette of the aircraft from the base of the tower. Tragically in so doing the very reduced forward visibility meant he did not see the crane boom - which frankly was hard to see against the background in good visibility.
So I suspect that (ultimately) pressure to comply with the rules contributed to this accident. Though admittedly he would probably not have been in that situation if he'd fully complied with the rules from the outset.