American Prairie in Montana is an example going the other way: a nonprofit buying private land that surround public land and opening them to the public. It's somewhat contentious locally, but it shows how much the private parcels around public land shape who can actually use it.
How does the typical suburban family look like? Did all 3-4-5 of you drive F150s there?
Other people deserve access to public land too but might not pass the vibe check of the private guards, which is what op was lamenting as well.
Well, that depends on how large and obvious your property is.
Assuming you have a typical suburban 1/2 acre or 1 acre property, the simplest thing to do is put up a fence, or otherwise make sure that there are no trails that go over the property you use.
Otherwise, you need some kind of bounds marking and obvious "no trespassing" signs.
FWIW: The site identified public lands near my house that have a lot of trails on it. There are trails that go right behind peoples' yards, and most of them either have fences or sheds that make it obvious where the bounds are.