  Motivated by experiments on the archetypal frustrated magnet
SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ (SCGO), we study the classical Heisenberg model
on the pyrochlore slab (Kagom\'e bilayer) lattice with site-dilution $x=1-p$.
This allows us to address generic aspects of the physics of non-magnetic
vacancies in a classical spin liquid. We explicitly demonstrate that the pure
($x=0$) system remains a spin-liquid down to the lowest temperatures, with an
unusual {\em non-monotonic} temperature dependence of the susceptibility, which
even turns diamagnetic for the apical spins between the two kagome layers. For
$x> 0$ but small, the low temperature magnetic response of the system is most
naturally described in terms of the properties of spatially extended spin
textures that cloak an "orphan" $S=3/2$ Cr$^{3+}$ spin in direct proximity to a
pair of missing sites belonging to the same triangular simplex. In the $T
\rightarrow 0$ limit, these orphan-texture complexes each carry a net
magnetization that is exactly half the magnetic moment of an individual spin of
the undiluted system. Furthermore, we demonstrate that they interact via an
entropic {\em temperature dependent} {\em pair-wise exchange interaction}
$J_{eff}(T,\vec{r}) \sim T {\mathcal J} (\vec{r} \sqrt{T})$ that has a
logarithmic form at short-distances and decays exponentially beyond a thermal
correlation length $\xi(T) \sim 1/\sqrt{T}$. The sign of $J_{eff}$ depends on
whether the two orphan spins belong to the same Kagome layer or not. We provide
a detailed analytical account of these properties using an effective field
theory approach specifically tailored for the problem at hand. These results
are in quantitative agreement with large-scale Monte Carlo numerics.
