Disease was a really big problem.
The history suggests there was far too much of it for gardens to make much of an impact in it. There was also a lot of horse urine to deal with too.
There are a bunch of problems with cars, but I'd much rather live in one of those cities as they are today than with no cars but mountains of horse manure everywhere.
Modern New York frequently has mountains of unremoved waste whenever there's a sanitation dispute .. it's a problem that hasn't been solved.
New York and London of yore had logistical challenges that could have been improved, London famously rebuilt its sewers to address the the miasma, and there are many uses for urine, horse or human, if gathered.
It's more an infrastructure issue, dealing with waste, than an intrinsic failing of one mode of transport over another.
It is exactly an intrinsic failing of one mode of transport over another, that it produces unsanitary biological waste which at scale makes life pretty unpleasant in big cities. "It's an infrastructure issue" doesn't help if the infrastructure to solve the problem didn't exist and wasn't getting built (London's sewer upgrade was built some time before the horse situation was considered critical and was primarily to address problems with human effluent).
CO2 isn't unsanitary, although it is biological waste (some millenia removed from its time of origin) and the first major London Sewerage upgrade was mentioned as an example of building infrastructure at scale not as an example of removing horseshit .. that situation wasn't seen as serious enough to address with a dedicated grand scale service prior to ceasing to be a problem as horses went away.
For many decades the lead additives in petrol met the unsanitary definition (second clause) being "unhealthy and therefore likely to cause disease" withoiut being biological.
It appears, to myself at least, that "exactly" is less clear cut than you make out; no major effort was made to address horse waste (past the daily sit carts and shovels) and petrol, rubber particles, noise, increased speeds, etc come with a new set of problems which have still not been addressed.
There are a bunch of problems with cars, but I'd much rather live in one of those cities as they are today than with no cars but mountains of horse manure everywhere.