Seems like a cultural problem, not a crosswalk light problem. Most people (myself included) automatically slow down when driving into a crosswalk. Our intersections are usually fairly small, so it's hard to build up any speed, even if you wanted to.
It isn't every car by any means, but enough cars that pedestrians need to assume that drivers are incompetent until given evidence otherwise.
Strange, because most likely the majority of drivers also walk at least a short distance around the same cities they drive into.
In the US in particular, eyballing the number of cars and the number of pedestrians in images of places like New York etc that I've seen, it's very hard to believe that there's a clear line separating all those people walking about from all those people who drive cars, other than the specific time of day they do one or the other.
So you'd think that being careful around pedestrians should come naturally to most drivers, if nothing else because they'd want other drivers to be careful around them as pedestrians also.
You sound like a non-American. Let me give you some advice: take everything you've ever seen about NYC, and file that in a different place in your brain's filing system away from your file labeled "America". It does not represent America in any way when talking about cars, walkability, culture, or really anything else really. It's a place that's totally unique in the world. For some reason, too many non-Americans see movies set in NYC and think that all of America is just like this, and it isn't at all.
(Personally, I wish it was more representative of America in terms of walkability, but it just isn't.)
(This is not to imply that severe injury and death are appropriate "punishments" for walking inattentively, merely that the pedestrians have large incentive to modulate their behavior, while drivers do not.)
But it certainly seems true in busier areas with less pedestrian traffic such as mini malls. Especially when pulling out of parking lots.
This is pretty limited to urban areas, in my experience. In more suburban and rural parts of the country, people are much more cautious and obedient to traffic signals and rights of way. Of course there are exceptions in both cases, but running red lights, turning into pedestrians/cyclists/etc is much more prominent in major cities. As an urban pedestrian, I just have to keep my head on a swivel because there's a good chance that the taxi/Uber across from me is going to run the light that just turned red so he can make his left turn instead of waiting for his signal.
Right-on-red is a huge problem in areas where pedestrians want to cross. If memory serves, the UK does not allow turns on red. That makes more sense to me.
On my walk to work, I have to cross one 8-lane suburban street. 4 thru lanes, plus several turn lanes in each direction. The radius of the corners is smooth. Stop lines are far back from the actual apex of the corner. So, cars can easily roll through corners at 40mph. They do this when I have a walk signal because they either have their own green, or they're turning on red and only looking for oncoming traffic (to their left) and not for pedestrians (to their right).