The eigenvalues become relevant once you employ some sort of the quantum correspondence principle, like Weyl quantization. In such cases, the requirement is that the results of measuring the same observable (like energy) should be comparable in quantum and classical physics, and a part of that is agreeing in magnitude and dimension. It does not affect probabilities as values, but affects what events the probabilities are assigned to.
For an observable like Pauli Z in quantum information this indeed does not affect much, and the interpretation or realization of the measurement does not change if we take any observable of the form $a + b\sigma_z$ instead. But if we speak of a spin-Z measurement, it must be of the form
$$S_z = \frac ħ2σ_z$$
and give values of $+ħ/2$ ($|\!↑〉$) or $-ħ/2$ ($|\!↓〉$) with the dimension of action, so that it can be directly compared with other angular momenta.
If correspondence to real world measurements is not important to you, there have been some attempts at relaxing the Hermiticity condition, whose main use is to guarantee a real-valued spectrum. This relatively popular article, for example, suggests using complex units instead for observables which are naturally 2π-periodic (thus describing measurements using unitary operators instead of Hermitian), and goes as far as claiming that not even normality is necessary.
If you only are interested in the probabilities, and are happy to assign them to abstract events, you can ditch the operator observables entirely and introduce some more general concept like positive operator-valued measurements instead, which do not describe the measurement using any single operator. I think it's also done in your book, just somewhat later.