Asymptotics of the number of words in a regular language of given length
The argument you have sketched appears to be in line with Richard Stanley's treatment of the Transfer-Matrix Method in Enumerative Combinatorics, Volume 1 (link: pp 573; print: pp 500).

He starts with the generating function, and unpacks it by considering digraphs and permissible and prohibited factors. He then abstracts to free monoids, where he uses a refined version of the sums you gave to prove:


  4.7.11 Proposition Let $B$ be a subset of $A^*$ that freely generates $B$. Then $B^*(\lambda)=(I-B(\lambda))^{-1}$


After working through some applications, he likewise closes the section by discussing Hadamard products in relation to horizontally-convex polyominoes.