We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum.
Is there any similarly simple system which we need to describe using a (two-component Weyl) spinor?
Yes, there is a simple system which is efficiently described using 2-component spinors: the dynamics of a classical spinning top.
As Andrew Steane says, in his excellent undergrad introduction to spinors https://arxiv.org/abs/1312.3824
“It appears that [Felix] Klein originally designed the spinor to simplify the treatment of the classical spinning top in 1897. The more thorough understanding of spinors as mathematical objects is credited to Elie Cartan in 1913. They are closely related to Hamilton’s quaternions (about 1845).”
Since one definition of a spinor is its 720-degree identity under rotation, Hamilton’s quaternions, used to describe e.g. rotations in computer imagery and aeronautics, can also be considered as spinors of a relatively simple, but classical kind.
For a more detailed discussion, see Chapter 5 of Laszlo Tisza’s MIT lectures on Applied Geometric Algebra ( https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009/lecture-notes-contents/Ch5.pdf )
“The two-component complex vectors are traditionally called spinors. We wish to show that they give rise to a wide range of applications. In fact we shall introduce the spinor concept as a natural answer to a problem that arises in the context of rotational motion.”
Tisza’s chapter works through from triads to Cayley-Klein parameters and thence to 2-complex component spinors. Technically, his spinors would be non-relativistic Pauli spinors rather than Weyl spinors as they transform under SU(2) rather than SL(2,C).
The same topic (no pun intended) is touched on by David Hestenes in his New Foundations for Classical Mechanics (2nd ed. pp. 466+) and, in terms of Cayley-Klein parameters only, by Springborn in https://arxiv.org/pdf/math/0007206.pdf (Goldstein, as advised by Alex Nelson above, mentions spinors in his 2nd edn (1980); but not in his 3rd edn (2002), nor, according to Hestenes in the 1st edn 1950 or 1951).
Late addition: David Hestenes paper Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics addresses the OP's question more directly.
"A wider use of spinors in classical mechanics ought to dispel the pervasive mistaken impression that spinors are a special feature of quantum mechanics."
But it should be borne in mind that Hestenes' definition of spinor rests on an interpretation of the i in the usual complex plane as a bivector, enabling him to label the complex plane as the "spinor i-plane". This interpretation is elaborated in 'Section 2-2. The Algebra of a Euclidean Plane' in his New Foundations for Classical Mechanics (op. cit.).