Why is "points exist" not an axiom in geometry? I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.
 A: In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.
A: Getting a bit meta here, but the point of axioms is not just to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.
Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds".
However in maths it is not clear that would always be the case.
Indeed there are interesting cases in which it doesn't hold, and these need to be differentiated.
EDIT below, some extra thoughts:
Often the 'existence' axioms are not needed for the simple grammatical reasons. It's often the case that arguments go (or can be rephrased) something along the lines of: the following restrictions exist/all things considered must obey these rules/there are no things that don't obey these rules. If this is a true statement without the caveat of non-emptiness, then it is in the general case as the empty case is automatically true. This is often the easier to work, and doesn't need 'points exist' as an axiom.
A: One point that is missing from the other answers is that of vacuous truth. If formulated properly, we do not need points for the theorems in geometry to hold. If there are no right angled triangles then the Pythagorean theorem trivially holds for all of them, if there are no circles, Thales' theorem hold for all of them and so on. There might be a few theorems that cannot be restated in the form "For all objects x of class Y we have that ...", but even those could be reformulated as "If there exists a point, then...".
So, say you take a model-theoretic point of view of geometry and look at it just as a "set of theorems that hold for any model which satisfies those axioms". Then adding "points exists" as an axiom will only exclude the empty set as a model, which was kind of a boring model anyway. There might be a reason to do so for convenience and readability, in the same way as for example most authors exclude $\{0\}$ from the fields, but the differences in the resulting theory will be rather minor.
