A little bit. My humor does tend to skew a little dark, most of the time; and some of that did filter into the response. There’s also a legitimate observation about Batman in there, and it gets back to the fantasy surrounding the character.
Anyway, if I’m remembering correctly, you are correct. At a policy level capital punishment doesn’t work as a deterrent. There’s probably a joke to be had about it “doing wonders for recidivism,” but ultimately, it’s just too rare and unpredictable to function as a disincentive.
As a statement about Batman’s philosophy and state of mind? It’s a legitimate assessment of his position. Mr. “Criminals are a superstitious cowardly lot” believes that extrajudicial violence is just what Gotham City needs. More than that, it has to be his extrajudicial violence. Within that specific context, accidental deaths fall into the range of, “eh, close enough.” Again, if we ignore his personal code against killing people.
That said, it doesn’t seem to work. Gotham is a hellhole, even with (and arguably because of) his activities. A fact which seems to sail right over his dubiously themed cowl.
There’s a legitimate (and fairly well supported) read of the character, that he actually causes more harm than good to Gotham; that his behavior encourages the rash of supervillains who plague the city; and that he is just as crazy (if not more so) than the people he tosses in Arkham.
In case it’s not apparent, this isn’t automatically a bad thing. When handled well it’s one of the more interesting aspects to the character. It’s also worth keeping in mind for your own writing. You can have characters who hold beliefs that do not align with objective reality and insist that they’re right and the world is wrong.
Also, people can suffer from confirmation bias (the thought process where you subconsciously weigh evidence based on how well it supports your expectations), so it’s not like this element of the character is unrealistic. He’s already so far around the bend that dressing up as an airborne rodent seems like the best way to deal with the city’s crime problem. That isn’t exactly the kind of person you’d expect to provide well reasoned policy analysis and tailor their response accordingly. That he can survive night to night beating the hell out of people? That’s not so realistic, but the part where he believes in his cause; evidence be damned? That happens.