I actually think Miller's caption boxes/inner monologue are a lift from prose rather than comics. They're his throwback to old-time pulp fiction because they're not written as normal comic book thought balloons. Comics is a print medium, so I always thought his innovation was an interesting way to mix the strengths of prose into the form of comics to gain benefits from both. The only one who mixed prose and comics better was Dave Sim in Jaka's Story, where he really exploits the past-tense nature of most prose writing vs. the present-tense nature of comics (and, it should be pointed out, of film, which becomes important in a minute).
As I pointed out in my review of the first film, the problem with turning Miller's caption boxes into voice over in movies is that they become the equivalent of badly used thought balloons in comic books -- overly expository and betraying a lack of trust that the story points you need to communicate can be carried by the language of film. I think that's been a major stumbling point for every other Frank Miller filmed adaptation to date (even Batman: Year One, which was the best of them before now IMO).
I think this is the also root behind the loss in texture and depth in the movie vs. the book (and in moving from prose to film in general) -- you can go into a lot of more depth in prose without losing the audience the way you will if you stop a movie dead to have someone expound on their line of thinking or comment on the meaning behind events. Prose can essentially freeze time at will -- someone can take pages to talk you through a thought process you whip through in microseconds in real life and it'll work if it's written well enough. Prose is usually past-tense. Trying to do the same in film directly means everything else has to stop so you can talk talk talk about stuff, and that's death for a film. Film is happening in front of you, so it's fundamentally present-tense. You lose way too much momentum if you leave in lines like, "Why do you think I wear a target on my chest?" or even "20 million die by fire if I am weak," as much as I like both of those lines in the book.
The fun part is how comics can control your perception of time in a different way than straight prose through the use of panel shape and size and other tricks. Read Miller's work again and pay attention to how he uses panel sizes to make you slow down at key moments, vs. the bigger panels that will make things read faster. He's one of the best in the biz at doing that, and in a very specific way to evoke an emotional response. You have to go to Will Eisner or Osamu Tezuka for someone comparable.