They aren't really special except that adjoining the solutions to a radical to a field make the associated group of automorphisms simplify in a nice way. Also at the time tables of radical roots where common technology and Newton's method was not. But fundamentally, we can define and use new functions to solve things pretty easily; if you get down to it cos() and sin() are an example of this happening. All those applied maths weird functions (Bessel, Gamma, etc) are also this. As well, the reason not to just use numeric solutions for everything is that there are structural or dynamic properties of the solutions of equations that you won't be able to understand from a purely numeric perspective.

I think taking a specific un-solvable numeric equation and deriving useful qualitative characteristics is a useful thing to try. You have cool simple results like Lyapunov stability criterion or signs of the eigenvalues around a singularity, and can numerically determine that a system of equations will have such and such long term behavior (or the tests can be inconclusive because the numerical values are just on the threshold between different behaviors.

That's one of the really fun things about taking Galois theory class - you get general results for "all quintics" or "all quadratics" but also you can take specific polynomials and (sometimes) get concrete results (solvable by radicals, but also complex vs. real roots, etc.).