  In this paper, we introduce two new approximation properties for \'etale
groupoids, almost elementariness and (ubiquitous) fiberwise amenability,
inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show
that, in their respective scopes of applicability, both notions of almost
finiteness are equivalent to the conjunction of our two properties. These new
properties stem from viewing \'etale groupoids as coarse geometric objects in
the spirit of geometric group theory. Fiberwise amenability is a coarse
geometric property of \'etale groupoids that is closely related to the
existence of invariant measures on unit spaces and corresponds to the
amenability of the acting group in a transformation groupoid. Almost
elementariness may be viewed as a better dynamical analogue of the regularity
properties of C*-algebras than almost finiteness, since, unlike the latter, the
former may also be applied to the purely infinite case. To support this
analogy, we show almost elementary minimal groupoids give rise to tracially
Z-stable reduced groupoid C*-algebras. In particular, the reduced C*-algebras
of second countable amenable almost finite groupoids in Matui's sense are
Z-stable.
