Algebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.
Even before Grothendieck's scheme theory, algebraic geometry was already pretty focused on varieties in rings of polynomials and other such constructions.
The heart of the domain is still using abstract algebraic arguments to solve geometric problems.
E.g.; Euclid's method for finding the midpoint of a line is to draw two concentric circles centred at the vertices with radius the length of the line. The straight line that passes through the two intersection points of the circles, also passes through the midpoint of the line.
This is the same as saying the midpoint of a line is the intersection of an algebraic variety with a root at one vertex, and another algebraic variety with a root at the other vertex.
You don't need schemes, or projective curves, or local rings to prove it.
