Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?

While reading a paper (pdf) about the history of modern logic, I learned that some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to Moritz Pasch. After googling for Moritz Pasch, I was surprised to learn that he had found important implicit assumptions in Euclid missing from the axioms/postulates. I read on wikipedia that both Pasch's theorem and Pasch's axiom cannot be derived from Euclid's postulates.

Are there simple models similar to elliptic and hyperbolic geometry for the parallel postulate that allow to illustrate this fact in a simple way?

There are no simple models. To violate Pasch's Axiom in a Hilbert-type setup, we need to use a discontinuous solution of the Cauchy functional equation $f(x+y)=f(x)+f(y)$. Such a discontinuous solution requires (part of) the Axiom of Choice.

In ZF with added axiom that every set of reals is Lebesgue measurable, the Pasch Axiom is a theorem of a Hilbert-style axiomatization that leaves out the Pasch Axiom.

Remark: The first construction of a non-Paschian geometry that otherwise satisfies the full set of Hilbert's axioms is due to Szmielew. A proof that such a geometry must be of the Szmielew type was given by Adler.