Can we view higher homotopy groups as symmetries?

If we want to think of a group as representing symmetry, we want it to act on something. In the case of the fundamental group of a (nice) topological space $$X$$, even though it's definition doesn't a priori have to do with symmetry, it turns out that $$\pi_1(X)$$ acts faithfully on the universal cover of $$X$$ by deck transformations. So even though $$\pi_1(X)$$ isn't the symmetries of $$X$$ itself, it is the symmetries of a construction related to $$X$$.

Are there any natural actions of higher homotopy groups that let us view them as symmetries of something related to $$X$$?

(Of course, every group acts on its underlying set, but I'm wondering if there's a less tautological action that's actually related to the topology of $$X$$.)

All of the higher homotopy groups are also fundamental groups, so your construction with the universal cover also applies to them. Namely,

$$\pi_2(X) \cong \pi_1(\Omega X)$$

where $$\Omega X$$ is the based loop space of $$X$$ (at the same basepoint at which we're taking homotopy groups, which is being suppressed in this notation), so $$\pi_2(X)$$ also acts by symmetries, on the universal cover of $$\Omega X$$. This universal cover, when interpreted directly in terms of $$X$$, is (up to homotopy) the based loop space of the second stage of the Whitehead tower of $$X$$. This is a kind of "higher universal cover" of $$X$$ which trivializes its first two homotopy groups. (However, unlike the universal cover itself, it doesn't have a nice point-set description in general, e.g. it is not guaranteed to be a manifold if $$X$$ is a manifold or anything like that; it is really a homotopy-theoretic construction.)

Similarly $$\pi_n(X) \cong \pi_1(\Omega^{n-1} X)$$ and the same applies here.