I don't have a crossing nearby to check but I would expect there will be a period in the control system when both cars and pedestrians are shown a stop signal.
As a driver, I would be looking for people who are waiting, if so I have to assume the crossing may be about to change.
As a pedestrian, even if the crossing is telling me to cross, I still check to see the road is clear.
Blind people, make much more use of hearing than sighted people, that's how they will check the road is clear before moving off.
As has mentioned above, the law is absolute, but as in most laws there are grey areas to be adjudicated in court.
As traffic has got denser, more drivers are just following the car in front, increasing the chances of them being on or very near the stop line as the signal changes, they then are surprised and and fail to stop.
They have broken the law but I would rather wait on the footpath than get in their way!
From the OP's description, I suspect that this is NOT a pedestrian crossing (Pelican, Toucan, etc), but a pedestruian phase in the junction lights system
A big difference is that the latter does not have a stop line. The only stop line that the driver has to obey is the one on the approach to the junction. If he crossed that on green, no traffic light offence was committed.
Having said that, he has a duty of care towards the peds, and could commit careless or even dangerous driving. Conversely, the pedestrian should ascertain it is safe before starting to crossi, though it seems few actually do.