I think the construction you're looking for can be seen as a right adjoint, and hence the details of the construction can be seen as coming from general transfinite constructions of adjoints.
$\newcommand{\inl}{\mathrm{inl}} \newcommand{\Coalg}{\mathbf{Coalg}}$ There's a functor $\inl^* : F$-$\Coalg \longrightarrow (F+1)$-$\Coalg$; it embeds $F$-coalgebras asthe full subcategory of "error-free" $F+1$-coalgebras, and is induced by the natural transformation $\inl : F \rightarrow F+1$ in an obvious-once-you-write-down-the-diagram way.
Now, if I'm understanding right, the construction you're looking at, the "error-free core" of an $F+1$ coalgebra, is the right adjoint to this.
Moreover, I think there should be theorems that show automagically why this can be computed by the construction you give, as an $\omega$-long limit of pullbacks — but I'm not sure exactly where, I'm afraid. It's almost certainly deducible from the Kelly "Unified treatment of transfinite constructions" paper, well-described by Tom Leinster here; the constructions of that have a very similar flavour.
Relevant
Possibly relevant well-known constructions to compare (in Kelly and elsewhere): the construction of an algebraically-(co)free (co)monad on an endofunctor, and ; the construction of the free $T$-algebra on a $T$-graph.T$-graph; the free $S$-algebra on a $T$-algebra, given a monad map $S \to T$.
