But the question was different: it wasn't "can I get good at this flowery small talk if it doesn't come naturally?", it was "is flowery small talk genuine care?"

I would posit that no, it is not. And it's not even unambiguously a good thing. There are plenty of cultures where people are described as cold until you get to know them, but once you do - they'd die for you. To me, that is genuine care. The American "Hiii! How ARE you? I don't actually care if you keel over and die!" approach feels fake.