This is not true and there are numerous counter-examples.
City streets have always been dangerous for pedestrians and notable illustrations of this are the separate gates at city entrances for vehicles (pulled by beasts of burden of course) and pedestrians and the construction of sidewalks and crosswalks one example of which I've seen in person is the civil engineering of Pompeii where pedestrians were segregated from road traffic over two thousand years ago. https://commons.wikimedia.org/wiki/File:Pedestrian_crossing_...
Another example is the Pont Neuf bridge in Paris. Completed in the early 1600s it was built with wide sidewalks to keep pedestrians out of the road. https://upload.wikimedia.org/wikipedia/commons/b/b9/P1140241...
Oldest photograph of London: https://londonist.com/london/art-and-photography/oldest-phot...
Amsterdam, 1891: https://en.m.wikipedia.org/wiki/File:Old-Amsterdam_1891-stre...
What do you see? Sidewalks.
In various paintings and engravings of city scenes from the 1600s and 1700s a common theme emerges: every area that could afford them built sidewalks and pedestrians were segregated from road traffic. Raucous examples of people teeming in the streets in similar works seem to be trying to depict scenes of anarchy or poor slums.
Another is the fact that numerous Roman emperors banned carriages from city streets during daylight (both for safety and noise pollution reasons) with varying degrees of success. That was much easier when you had an army of slaves to carry goods into the city.
One glaring expression of the dangers of pre-automobile urban streets is the fact that in New York City fatalities due to horse-pedestrian collisions greatly surpass the rate of fatalities of automobile collisions. In both "Herald Square, 1896" and "A Trip Down Market Street, 1906" you can see pedestrians travelling with the road along sidewalks and crossing the road at any point. This indeed used to be common, before the invention of traffic signals and crosswalk and is the primary reason for the increase in fatality rates due to collisions.
The fatality rate for deaths due to horses in Chicago was seven times the present rate of pedestrian fatalities and in New York City is was double.
From Horse Power to Horsepower: https://escholarship.org/uc/item/6sm968t2
Herald Square: https://www.youtube.com/watch?v=PRf5z75-GbU
Market Street: https://www.youtube.com/watch?v=8Q5Nur642BU
Children didn't play in city streets because it was a death sentence.
Pedestrians DID cross from sidewalk to sidewalk at any location because crosswalks didn't exist (in most places) yet.
Regardless of all of that, in positive news more and more cities all over the US and the world are pedestrianizing their streets. Not because "users of city streets were diverse and included children at play and pedestrians at large" but because mixing pedestrians and ANY vehicle to include pre-automobile vehicles is dangerous and pedestrians should either stay out of the street or have it dedicated solely to their use.