Well, force yourself to care.
I'd argue that there is a very strong value in doing something good, not just because it's genetically or socially imprinted on you, but because you actually decide to do it.
This applies to everything, there is no merit in being good at something just because you were born that way.
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.