The principal use of probability amplitudes is as the physical meaning of the wavefunction, a link first proposed by Max Born and a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the wave function were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation was offered. Born was awarded half of the 1954 Nobel Prize in physics for this understanding, though it was vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein.Probability was just Born's interpretation, and not essential to the Copenhagen interpretation.
I have two simple reasons for saying that quantum probability is just an interpretation.
If an electron wave function tells you the probability of the electron's location, then you are implicitly assuming that the election is a particle. But the electron is not like a classical particle, and thinking of it as being like a particle is just an interpretation.
If the theory only predicts probabilities, then it cannot predict individual events. We cannot observe probabilities directly, so we can only confirm them by redoing experiments many times. Such an interpretation is weaker than one which predicts individual events, such as the positivist interpretation.
So I say that quantum mechanics needs no probabilities, not even for the double slit.