Yes and no, it's a bit too simplistic and doesn't explain the actual "why" of Galois Theory, just the how. The brilliant insight Galois figured out is that there is a fundamental connection between fields and groups, but that is just the "technique" with which he solved the problem. The "why even bother" is a bit more complex but simply put Galois wanted to establish a criterion to determine what polynomials are solvable or unsolvable in which fields. E.g. we know x^2=-1 is solvable in C with x=i but not in real numbers. Can we generalize that proof to such a degree that we can mechanistically run it for arbitrary polynomials in arbitrary fields?