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)