My parents, who live in Trinidad, have this problem every time they visit me in America. After a couple of days of everyone fake-smiling at them the whole time they get pissed off. To them, everyone's smiles come off as insincere and even slightly condescending. (They hate it even more when people tell them to "have a nice day".)
I'm an American that smiles and greets strangers regularly. It's not fake when I do it, and I know it's not fake when many of my acquaintances do it.
I do know that this is not a natural/universal thing (though it took a trip to Japan for me to materially understand that). But just because it's not universal doesn't mean that, where it happens, it's backed by some kind of underhanded, guarded cynicism.
So have a nice day. ;)