I don't think anyone disputes that the classical definition of particle and wave don't really apply in the quantum mechanics level. But QM makes good predictions. If QM talked about the blicket/fand distinction, and said that blicket was sort of like particle, and fand was sort of like wave, would you be more comfortable with it?

Because QM is the only scientific theory that explains observations, including the weird ones. That's something that needs to be acknowledged. The idea that math can't be used to describe reality is just a more specific way of saying that we can't describe reality at all.