Actually, "betweenness" can be defined from Hilbert's other primitive notions, in two steps:
1. For points $A, B, C$ say that "$A$ is closer to $B$ than to $C$", notated $AB\le AC$, iff for every line $\ell$ that contains $B$, there is a point $X$ on $\ell$ such that $AX=AC$.
2. For points $A, B, C$ say that "$B$ is between $A$ and $C$" iff the points are collinear, and $AB\le AC$ and $CB\le CA$.
Hilbert could have formulated his axioms based on these definitions (viewed as abbreviations) instead of making "between" a primitive notion. However, he probably wouldn't have considered that an improvement. He probably wasn't particularly focused on minimizing the number of primitive notions at any cost, and there would have been several costs:
The axioms themselves would be more complex to state, and less clearly true about pre-formal intuitive geometry.
It might be desirable/interesting to be able to cordon off the part of the theory that is invariant under (nonsingular) affine transformations of space -- but that can't be done if betweenness (which is an affine concept) were dependent on congruence (which isn't).
Geometry in one dimension would not be a matter of simply restricting one's attention to a single line and points on it, because the definition of $AB\le AC$ depends on the possibility that $\ell$ may be different from the line $AB$.
It would be unclear whether Hilbert's somewhat wonky "completeness" axiom (which basically asserts that we only want to consider maximal models of the other axioms) would work in this setting. At least a priori one could imagine a model that declared $A$ to be closer than $B$ than to $C$ simply because one of the lines through $B$ had gaps in it that allowed it to pass through the circle $AC$ without actually intersecting -- and this we could have deeply nonstandard models that were nonetheless maximal.
(Edit: Wikipedia's article on Tarski's geometry suggests the following simpler drop-in replacements for the definitions above, which avoid speaking about lines at all:
1. For points $A, B, C$ say that $AB\le AC$ iff for every point $Z$ such that $AZ=CZ$, there is an $X$ such that $AX=BX=CZ$.
2. For points $A, B, C$ say that "$B$ is between $A$ and $C$" iff $AX\le AB$ and $CX\le CB$ implies that $X$ must be the point $B$.
The costs above would still hold for these.)