Why aren't Grobner bases more common?
Well, I don't know what you expect to get as an answer, but in the comments you already got that if you haven't seen them used is due to two reasons.

The main one is probably that maybe you haven't had the need to solve problems in which they are needed, yet, and
that the problems, like finding a solution of a system of polynomial equations. are more efficiently done by algorithms that target exactly that problem, like Newton's method. Groebner bases can end up with large degrees for systems with relatively few variables and degrees larger than one.


The usefulness of Groebner bases becomes clear once you study what they are. They are  a form in which to put a system of equations. The form is canonical once you fix a way to order monomials in several variables. If you have seen some commutative algebra, Groebner bases are a canonical set of generators of ideals in a polynomial ring $I\subset R=K[x_1,x_2,\ldots,x_n]$.
When you choose a convenient monomial order it happens that the form has the same property as the row echelon form for linear system of equations, that the equations have less and less variables until eventually (for a system that has finitely many solutions) the last equation is a polynomial in one variables. So, incidentally you get out of it a method to find all solutions by backward substitution. You find all solutions of this polynomial in one variable, by either Newton's method, or perhaps some method that finds them all roots at once. This gives you the possible values of one of the variables. Then you substitute these values in the previous equations and repeat the process with them. See here for one way to do these steps.
Example: For the system $$\begin{align}x^2+y^2-1&amp;=0\\xy-4&amp;=0\end{align}$$
you get $$\begin{align}4x+y^3-y&amp;=0\\y^4-y^2+16&amp;=0\end{align}$$
Try it with Singular, by inputting
// Define the ring R[x,y] of polynomials in variables x,y; 
// and lexicographical ordering to order monomials.;
ring r = 0,(x,y),lp; 
// The ideal generated by those two polynomials. ;
ideal I =(x^2+y^2-1,x*y-4);
// The same ideal, but now generated by its Groebner basis.;
ideal GI=groebner(I);
// Print it.;
GI;

As you see, Groebner basis is not an algorithm, but a form for a system, and its goal is not to find one solution, but to put the system in a form that helps compute properties of the set of solutions as a whole. Modulo technicalities, the set of solutions is called an algebraic variety. Problems that you might need to solve could be, for example,

What is the intersection of algebraic varieties? More explicitly, suppose you have a mechanical arm with both ends attached to two fixed points and having two or three elbows (joints). What are the positions that these elbows can be? It is an algebraic variety that can be described first as the intersection of the equations that you get by imposing the length of each segment of the arm and the condition on the fixed points at each end.

Parametrization of algebraic varieties. If the mechanical arm above had only one elbow, its positions would be a circle. Perhaps you know that $t\mapsto (\cos(t),\sin(t))$ parametrizes a circle, when $t\in[0,2\pi)$, but it could be better to get a rational parametrization, like $t\mapsto(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})$, for $t\in[-\infty,\infty]$. Such parametrizations always exist. You compute them by an algorithm that uses Groebner bases among other things.

Maybe you need the inverse problem of (2), implicitization. You want a robot arm that can cut this shape that the design team chose for the silhouette of a car. They sure like their Bezier curves. They happen to be algebraic.

You want your computer to discover the proofs of all synthetic geometry problems that you did in high school. After sufficient squaring, the hypotheses and the thesis are polynomial equations. So, you only need to check that the first can reduce the latter to $0=0$. This a particular case of problem (1).