How beneficial are Gröbner bases for solving systems of equations

There are several improvements you can make on Buchberger's Algorithm for computing Groebner bases, to the point where using other methods usually doesn't make sense. Most computer algebra systems use Groebner bases for their computations since they typically outperform any other method. I think one main reason Groebner bases are so revered is that they provide constructive solutions for many important problems. Take Hilbert's Syzygy Theorem for example. You can prove it using abstract nonsense homological algebra stuff, or you can prove it using Groebner bases. The Groebner basis approach actually gives a recipe for an algorithm to compute finite free resolutions, which is vastly more desirable.