The larger the city, the higher is the likelihood of accessible but nevertheless private property having those same things; though in either case its likely to be restricted to particular parts of the city, and those are likely to be the ones with lower property values (this reduced property value is both a cause and effect of the condition, its a positive feedback loop.)
The larger the city, also the more likely public -- and accessible private -- spaces are to be fairly pristine enclaves. These will usually be found in substantially different parts of the city than the nasty bits.
Besides they yard was an after thought, I spent most of my outdoor time in public places and I was outdoors every day in the summer.
You may just as well say "I'm too good for public property, I'd rather not share with the peons." Don't hide behind "safety." There's no major safety issues in letting your children use a public park.
And anyways, it's probably much safer to let your kids have a hypothetical chance of running into hypothetical broken glass (that they can just walk away from) than to put them in a car every time they need to go somewhere, as automobile accidents are the leading cause of death in people ages 5-25 (or something along those numbers)
I wouldn't call it public, it's more of a shared space an apartment complex might have.
The larger the city, the higher is the likelihood of public spaces having a variety of homeless encampments, broken glass, needles and plain old feces.