I'm not aware of any public schools in my area that would allow me to, e.g., use the basketball court or soccer field after school hours or on the weekends.
That probably says more about the area you live in than the public schools.
Around here the grounds are not only open outside of school hours, but explicitly so (they have closing hours posted: 9PM).
Have you tried? I've certainly been able to. And I'm definitely not alone in having used those facilities. I've used them personally and for ad-hoc sport events (lacrosse isn't exactly popular in the area I'm in right now).