Infinite density doesn't seem "weird", it is meaningless and indicates that our model is incomplete and simply cannot make predictions beyond a certain point. You don't look at an equation that divides by zero and think, "maybe someday this will make sense." It will never make sense because it will always be undefined, and you need to start looking for your mistake.

You seem to be suggesting that there is a non-classical physical model that resolves the paradox. But you don't claim that any such one exists. The physics inside a black hole might be different than here on Earth, but mathematics is not. There is no mathematical way to interpret the singularity, and so there can never be a physical interpretation. The model is meaningless deep inside a black hole. We will not know what happens until we develop a more complete model, not a better interpretation of this bogus prediction.