Axiomatic/synthetic geometry is the most intuitive way to introduce the concept of building up complicated ideas from simpler ones without relying on dense algebraic proofs which intimidate people—you just “construct” new objects
It’s historically super important; I think Euclid’s Elements is the most-read book after the Bible due to its role in historical education.
Through Hilbert, it helped found modern mathematics and abstract geometry.
It also lets us work ideas that may be difficult to express analytically in a simple, intuitive form.
I would recommend checking out the game Euclidia for an elegant approach to axiomatic synthetic geometry. It expresses the feeling of playing around with geometric objects to make new ones. This was ultimately the root of our modern proof-based approach to Mathematica.
Finally, most historical Western math up until the 1700s is grounded in axiomatic synthetic geometry. For example, Newton’s mathematic writing all rests on Euclidean geometry.