Yes we do (if you're a US citizen). You even get to vote for the "board of directors" so to speak (the legislative/executive branches of the US gov).
As for your other example, I can own quite a bit of a company and still get denied access to their offices and computer systems.
This high minded notion of "public owned" is quite different than what we are discussing here. The fact we're having this goofy conversation is sad.
I own part of Google. I can't take it home with me. I don't even know what that would mean.
> If I own it, I can get a permit to build on it.
It is indeed possible to get a permit to build on/use public grounds.
> If I own it, I can refuse to allow everyone else to use it.
You can vote for representation that could in theory restrict access to just you. The other owners probably wouldn't agree to it, though. There are a lot of owners who want open access.
I think you're confusing things here. You seem to be talking about having full ownership of something, as if partial ownership isn't a thing.
The public doesn't have a home, but if it did, it already took the national parks there. The public can get a permit to build there. The public can refuse to allow anyone else to use it. Remember: members of the public are not the public. "Anyone else" here actually includes members of the public (ie: you and I).