"Trespassing" on public lands - a bogus concept?

Here is something I have always had a problem with: the concept of "trespassing" on government property. I can somewhat understand this concept applied to specific context - say, it is reasonable to not let random people into the White House, which is essentially a residence of the president and his/her stuff. However, it seems that nowadays governments around the world arbitrarily declare any lands they "own" to be closed for public.

I am an avid low-budget traveler, often staying for the night in my car on highway Rest Areas during my trips. On my last trip to Everglades I stayed on a Service Plaza in Miami (Service Plazas are essentially fancier Rest Areas, featuring some shopping opportunities, gas stations, etc.). In the morning, doing my usual running-exercising routine, I had a police officer drive my me and yell in megaphone: "Sir, you are not allowed to do it here!" When I inquired as to what law it was I was breaking, he said, "This is government property. You are only allowed to stop here to take a break and fill up the gas. You cannot exercise here; if you do, you are trespassing".

I thought it was utterly ridiculous, and next morning I was back there, doing the same set of exercises, this time only having one officer approach me and ask what I was doing and why, leaving me alone afterwards. But a greater implication came to my mind: is it at all reasonable to deny free citizens the right to exercise on a large chunk of land "owned" by the government, with free entrance to it?

"Governmental land" is supported with our taxes; should it not be our right to roam about it however we please, as long as it does not directly interfere with the responsibilities of public servants? Why should the government be able to say, "This land here is for the recreational use of public servants only", when it is private citizens who are funding this land in the first place?