Pedestrian safety is improved very very dramatically by cutting speed limits on urban streets to 25 or even 20 mph.

A pedestrian hit at 20 mph is likely to sustain minor injuries. A pedestrian hit at 40 mph almost always dies.

Drivers also have much better road awareness and much more time to react at 20 mph, and pedestrians likewise have a better chance of jumping out of the way of a slower-moving car.

One good street design idea is narrowing traffic lanes. This makes drivers slower and more careful, but doesn’t cause any increase in accidents, and doesn’t substantially reduce car throughput. It also provides extra space that can be used for bike lanes, sidewalks, and better designed intersections.

Well-designed roads for pedestrians are designed to limit traffic speed. Common methods in Europe are to plant trees down both sides, to narrow the traffic lane and to have parked cars meaning traffic has to stop to let oncoming traffic pass etc.

Drivers are much more attentive when they're always looking all around for potential obstructions.

Speed limits are, in turn, influenced by the road's design.

No, you're missing the physical picture. Someone driving at 30mph over 20mph saves time only linearly, but to maintain that speed, the vehicle has to work much harder as air resistance, the major factor above 15mph, grows with the square.

As such, it makes sense to slow cars down to 20, even 15mph - the time loss (if there even is any, people driving in cities are just accelerating unnecessarily from light to light) very much makes up for the dramatic reduction in kinetic energy that endangers pedestrians and other road users.