cars + tires did represent a major public health breakthrough at the time, because cities were struggling under the weight of daily animal fecal matter and animals dropping dead in the streets.
https://en.wikipedia.org/wiki/Streetcars_in_North_America#Hi...
While streetcars may have made it easier to transport people, you still need to deliver goods to stores. There’s also the problem of having to wait for tracks to be built.
Horse droppings, urine, and carcasses were a major inspiration for the formation of New York City's sanitation department, including such features as standard uniforms (white as I recall, originally) to emphasize sanitation.
[B]y 1880 there were at least 150,000 horses in the city. Some of these provided transportation for people while others served to move freight from trains into and around the growing metropolis. At a rate of 22 pounds per horse per day, equine manure added up to millions of pounds each day and over a 100,000 tons per year (not to mention around 10 million gallons of urine).
99% Invisible Podcast, "The Big Crapple: NYC Transit Pollution from Horse Manure to Horseless Carriages": <https://99percentinvisible.org/article/cities-paved-dung-urb...>
The sanitation challenges caused by the 2.5 million lbs of manure daily and 60,000 gallons of urine from the City's more than 100,000 animals led to the First International Urban Planning Conference.
"The Great Horse Manure Crisis of 1894"
<https://themuseumwithoutwalls.org/Site?itemId=19747>
Tracked transport is efficient, but it is not flexible, and streetcars on rails are inherently limited in routes, as well as being subject to congestion (delays and stoppages tend to propagate).
Note that as of 1900, horses also consumed 25% of all grain production in the US. One reason the industrialised world switched to fossil fuels was that the limits of biofuels had already been reached.
(This is of course not the same as suggesting that fossil fuels are without their own side effects, limitations, or are inexhaustible.)
While they may have not been the main cause of the epidemics at the time, they certainly had some large part in it. The main part simply too many people packed too dense together in not so good conditions.
And the horses giving a shit...
In New York City in 1900 for example the pedestrian fatality rate from horse accidents was higher than the 2003 pedestrian fatality rate from cars [1].
Here's an comment from a medical historian with similar numbers for England and Wales [2]. About 70 road traffic deaths per million population from 1901 to 1905 compared with 80-100 in the 1980s and 1990s from cars. (It is down to around 26 nowadays).
We are better at treating injuries nowadays so it is likely that had they had modern medicine circa 1900 a fair number of those fatalities would have just been serious injuries.
[1] https://legallysociable.com/2012/09/07/figures-more-deaths-p...
[2] https://www.bmj.com/rapid-response/2011/10/31/cars-and-horse...
Unfortunately it’s going to take decades to fix the car mistake