To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the use of a "circle" (ignoring the Bloch sphere) to represent a combination of |0> and |1> states rather than, for example, a system with the condition that alpha+beta=1 (a line, in terms of its geometry). This still preserves the Complementary Rule if we consider the probability of |0> to be its coefficient alpha and |1> beta respectively. Is it just easier to use a circle, because the state can be represented in polar form in terms of some angle? If you had, for example, alpha+beta=1 or alpha^3+beta^3=1 the magnitude would also change but I don't necessarily see an issue with that because the magnitude doesn't need to represent the total probability of either event occurring (1).
So what is the fundamental issue with representing a quantum state using other conditions which still preserve the probability conditions? Does it have something to do with the magnitude? Linearity? Is it just convenient?