Was the notion of observable really necessary before quantum physics?

I’m a mathematician who’s been struggling with the search of connections between physics theories and $C^*$-algebras. The most known connection I found was that the observables in quantum mechanics are equivalently described by the elements of the $C^*$-algebra of self-adjoint operators on a Hilbert space, a result known as the Gelfand-Naimark-Segal construction.

On the other hand, I found some material on $C^*$-algebras that approaches this result by first understanding the notion of observable under the Hamiltonian formalism of classical mechanics. In this case, the observables are elements of the $C^*$-algebra $C_0(Gamma)$, where $Gamma$ corresponds to the phase space.

I personally found the notion of observable in the Hamiltonian formalism to be superfluous, because an observable being a continuous function means that we are able to do our measures with arbitrarilly small error, so, why not consider only the phase space itself? On the other hand, due to the uncertainty principle, it is a really necessary notion when it comes to quantum mechanics.

Therefore, I wonder, with the spirit of a beginner in this field, first of all, if the notion of observable in classical mechanics is really necessary and why so, and second, if there were other physics theories before quantum mechanics in which the role of this notion was as central as it is in quantum mechanics.