No, the basis for the modern treatment of Euclidean geometry is the explicit construction of the plane as R^2. Axiomatization is reserved for things like set theory and elementary number theory. Although Hilbert's axioms aren't exactly even a proper axiomatization anyway, seeing as it's a second-order axiomatization, and thus requires some ambient set theory. But if you've got set theory, you may as well construct it. You could think of the axioms as just conditions, of course, but then you need an existence proof, which is provided by the explicit construction as R^2, so...

Good point, my statement was poorly formulated. While I agree that analytic geometry and other developments in the field could be more fundamental, Hilbert's formalization can still be useful to express more precisely what one means by "points" and "lines", being, I think, applicable to the current discussion.

On the contrary, it's an axiomatization, and thus only describes the relation between them. If by "what one means by them" you mean a definition, then you'll need a construction for that. Of course, for the most part, the relation between them is what we mean by them; but Hilbert's formalism doesn't seem to be a good way to address such statements as the one that started this, "a point is a line segment of zero length".