Is this the normal understanding of private/public companies? I don't own the USPS nor do I own the national parks. However, taxes pay for them, so we are given access to them. NASA is another example. I can't just walk into the JPL and start using a computer. However, any data that is produced by NASA, I can use just like I can use the national parks.
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.
sidfthec is a little confused, but they've got the spirit. Here is the key: the public owns it. Not any individual member of the public.
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).