  Let $(X,g)$ be a compact manifold with boundary $M^n$ and $\sigma$ a defining
function of $M$. To these data, we associate conformally covariant polynomial
one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$.
They arise through a residue construction which generalizes an earlier
construction in the framework of Poincar\'e-Einstein metrics and are referred
to as residue families. The main ingredient of the definition of these families
are eigenfunctions of the Laplacian of the singular metric $\sigma^{-2}g$. We
prove that, if $\sigma$ is an approximate solution of a singular Yamabe
problem, i.e., if $\sigma^{-2}g$ has constant scalar curvature $-n(n+1)$ up to
a sufficiently small remainder, these families can be written as compositions
of certain degenerate Laplacians (Laplace-Robin operators). This shows that the
notions of extrinsic conformal Laplacians and extrinsic $Q$-curvature
introduced in recent works by Gover and Waldron naturally can be rephrased in
terms of residue families. This new spectral theoretical perspective allows
easy new proofs of results of Gover and Waldron. Moreover, it enables us to
relate the extrinsic conformal Laplacians and the critical extrinsic
$Q$-curvature to the scattering operator of the asymptotically hyperbolic
metric $\sigma^{-2}g$ extending work of Graham and Zworski. The relation to the
scattering operators implies that the exterior conformal Laplacians are
self-adjoint. We also derive new local holographic formulas for all extrinsic
$Q$-curvatures. Furthermore, we prove a new formula for the singular Yamabe
obstruction $B_n$. We use this formula for $B_n$ to derive explicit expressions
for the obstructions in low-order cases (confirming earlier results). Finally,
we relate the obstruction $B_n$ to the supercritical $Q$-curvature $Q_{n+1}$.
