I'm trying to understand/form an intuition for why a ket-vector, and its associated bra-vector, are appropriate tools to represent the spin of a qubit.
Generally speaking, if a system has a state/property, that state/property can be represented by a set of numbers that quantify the magnitude (and potentially direction) of that state/property. The magnitude is based on a scale (that hopefully has historical significance and/or has consensus of a large group of people).
Based on this understanding/logic, my intuition of the ket-vector representing qubit spin is that it in order to quantify spin of a qubit, we need 2 real numbers, which can mathematically be represented as a single complex number or a ket-vector. Thus, the ket-vector is just a mathematical tool that helps makes the math a little easier.
But what do the components of the ket-vector signify? Why do we need 2 numbers to specify a single property? I'm having a hard time coming up with other properties/states that need 2 numbers (except maybe waves which need amplitude and frequency to be fully defined - but then again, the properties of waves are amplitude and frequency which are themselves defined by single numbers).
I've read several articles/questions about the components representing the probability amplitude, and how $\alpha_u.\alpha_u^* = $ probability of finding that the spin of the qubit is aligned along $|u>$
This answer talks about how the probability theory used for Quantum Mechanics is slightly different than what we are generally used to, but does not provide any intuition as to why that's the case, and hand-waves it away by talking about interference (which I'm not yet familiar with).