Saying something =! communicated well. All Alfred says amounts to a conversation about the fact that something changed, not the "why" behind the change. Considering that the movie wants you to accept this Batman, it's a fatal mistake to show that Batman has seemingly changed without showing the reasoning behind his change.
Most of us can extrapolate that Robin's death has something to do with it, but as long as its never fully stated or shown, it's only a theory.
I would argue that extrapolation is misdirected. Robin's suit provides context to Batman's long history and loss. I don't think it has anything to do with him starting to care less about incidental murder. There's zero reference to Robin apart from the one shot. If it were recent, it would only make sense either of them would give it recognition. This leads me to believe the death was years back, and thus, has no bearing on his current actions now.
Again this goes back to the newspaper. Alfred questioning him indicates something has changed in Batman's follow-through. The newscast later on states that is the 2nd time Batman has done it. By all measures, branding is severely less of a punishment than murder. It's safe to say if Alfred starts questioning about branding, then manslaughter is still out of the equation. There's a blatant foreshadowing when Alfred says, "the feeling of powerlessness turns good men cruel". There's emphasis on that last part and clear disdain as he walks away.
The pier chase scene is our first glance at the kills. We know Bruce is after the Kryptonite. We know Bruce considers Superman a world-level threat and absolutely means to end him. Thus, that chase is for all intents and purposes a world-saving mission for Bruce. Enough cause for justifying some collateral death? In his head, I'd say so.
This is all set-up for Superman to shift Batman one way, and then shift him back (by the end). What are Bruce's last words? "There are still good men left" (callback to the newspaper Alfred scene). What is Batman's last action?
Not branding Lex Luthor.
Moving forward, I've great confidence Batman will be the modern one we know. No-kills is potentially back on the table.
With that logic you can justify anything this Batman does
That logic you refer to is exactly the one used to defend Nolan's Batman. Why is this laughable now? And do you genuinely consider it a "reach" to ask that question as a means of addressing the issue?
It would mean he's actively causing loss of life in a dumb mission he would later drop after hearing Superman say one word.
