Honestly, my answer to this came about many years after living through a very theoretical CS undergrad, by way of a chapter in Charles Petzold's book, 'Code' (of all things. I originally bought the book for my dad to help him understand 'what it was I did all day'). Basically, the insight centered around the physicality of how FA are taught; FA are usually taught by thinking of the FA itself as being fixed in space,…
It sounds a bit like you are talking about a Turing machine rather than a finite automaton (the latter is much weaker than the former in computational power). (Not that it matters much, your point still stands.)
I've seen the FA formalism used in computing theory a few times. Yet, I'm also missing whatever enlightment the author is looking for. But, if for no other reason, it's worth learning just because it's a very nice tool.