I can recommend an article Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg, The American Mathematical Monthly, Volume 117, Number 3, March 2010, pages 198-219. One of the great strengths of the article is that I am in it. Marvin promotes what he calls Aristotle's axiom, which rules out planes over arbitrary non-Archimedean fields without leaving the synthetic framework. If you email me I can send you a pdf.
EDIT: Alright, Marvin won an award for the article, which can be downloaded from the award announcement page GREENBERG. The award page, by itself, gives a pretty good response to the original question about the status of Euclid in the modern world.
As far as book length, there are the fourth edition of Marvin's book, Euclidean and Non-Euclidean Geometries, also Geometry: Euclid and Beyond by Robin Hartshorne. Hartshorne, in particular, takes a synthetic approach throughout, has a separate index showing where each proposition of Euclid appears, and so on.
Hilbert's book is available in English, Foundations of Geometry. He laid out a system but left it to others to fill in the details, notably Bachmann and Pejas. The high point of Hilbert is the "field of ends" in non-Euclidean geometry, wherein a hyperbolic plane gives rise to an ordered field $F$ defined purely by the axioms, and in turn the plane is isomorphic to, say, a Poincare disk model or upper half plane model in $F^{\; 2}.$ Perhaps this will be persuasive: from Hartshorne,
Recall that an end is an equivalence class of limiting parallel rays
Addition and multiplication of ends are defined entirely by geometric constructions; no animals are harmed and no numbers are used. In what amounts to an upper half plane model, what becomes the horizontal axis is isomorphic to the field of ends. This accords with our experience in the ordinary upper half plane, where geodesics are either vertical lines or semicircles with center on the horizontal axis. In particular, infinitely many geodesics "meet" at any given point on the horizontal axis.
Best Answer
To give a clear answer we need to define what you mean by synthetic geometry and analytic geometry.
I assume that we use the axioms of Hilbert to define synthetic geometry, and that analytic geometry corresponds to the real Cartesian plane $\mathbb{R}²$. If you assume only the first four groups of axioms (I-IV), you don't even have all the ruler and compass constructible points. There are propositions which are provable in $\mathbb{R}²$ but not using axioms I-IV. For example Euclid's proposition I 22 is not provable using only group I-IV. If you add an axiom about intersection of lines and circles then you get all ruler and compass geometry (see Chapter 10 of Franz Rothe's book).
But, if you assume the group V of Hilbert's axioms then the axiom system is categorical, the only model is the real Cartesian plane.
It is possible "to mimic analytic proofs in coordinates by constructing a coordinate system out of a synthetic geometry", it was first done by Descartes and with a more rigorous presentation by Hilbert in Foundations of Geometry (the proofs can be found in Chapter 4.21 of Hartshorne's book: Geometry: Euclid and beyond or Chapter 18 of Franz Rothe's book). The axiom of continuity (Dedekind and Archimedes) are not necessary for the arithmetization of geometry. Without continuity axioms, you obtain a Cartesian plane over a Pythagorean field.
It is not true that Hilbert's axioms does not have any sort of measurement, it is not explicit, but from Archimedes' axiom from Group V, it is possible to build a measurement function (see Chapter 8 of Franz Rothe's book).