Strikes me - and I may be very naive - that the nice thing about flying a helicopter is that you can 'stop' and hang around in one place for a bit while you contact ATC / wait thirty seconds for mist to clear / check a map. Or is there a fuel penalty for hovering? Increased workload when holding station? Is it just not done to hover in a lane?
I'm still puzzled as to why the a/c seems to have been approaching from the east -when the original flightpath must surely have taken it well to the west of London - and why the pilot did not contact Battersea control. If he didn't have time to communicate, suggests he was too busy aviating?
Your understanding of how helicopters are operated is flawed, in a number of ways, I'm afraid.
The relevant part about your first statement about "stopping". Hovering at altitude (outside ground effect) needs good visual references, full concentration with both hands on the controls, and considerably more power than slow flight. It's seldom done in good weather, let alone poor visibility, if flying from A to B. Police or other observation aircraft excepted, they need to stay in one place for over-riding reasons of the task in hand.
From the evidence already in the public domain, it appears that PB was flying a commonly used entry route to Battersea. He would have been directed to use that route because he would have been under radar control until he reported his position at, or in sight of, Vauxhall Bridge, a compulsory reporting point.
On reaching Vauxhall Bridge, he would have been required to turn almost 90 degrees right to fly westbound along the Thames, i.e. along helicopter route H4, to reach Battersea Heliport.
At that point, the normal thing is to be directed to change frequency to Battersea Tower because that is the eastern boundary of their ATZ and they control that airspace. It's not unusual to be given a maximum altitude of 1000 feet QNH at that point, because of other traffic above, descending onto the ILS at City Airport.
Just south of that point is the 770 foot high (slightly less now) building/ crane.
It seems to me there was nothing unusual about the circumstances leading up to the accident, apart from the deteriorating weather conditions.