Arrow - S8 E3 - Leap of Faith

The ultimate problem I have with that is, it's not for any lone person to take it upon themselves to decide that.  That's generally what it means to live in a society of rules; especially anywhere that has specifically lain out inherent rights to things like trial by jury and all of these other things that are there as some measure of measuring justice.  What gives Batman or Superman or any other lone individual the right to take it upon themselves, without any kind of sanction or authority other than their own person might and proactive will, to make those sort of extreme decisions?  It's one thing to disarm or apprehend someone doing harm or breaking the law, or protecting innocent lives in a time of crisis.  It's something else entirely to be judge, jury and executioner.
Admittedly, the conceit of someone like Bruce Wayne is that his city has become so corrupted, that his taking on the mantle of Batman is meant to be justified by him doing that in order to restore or instill into the city the virtues of justice that it lost, if it ever had them to begin with.  For him to act in any capacity that extends beyond that, and especially where he's executing people or taking such extreme measures as to lobotomize someone; he's no longer the advocate for justice in a corrupt system, he's become part of that corruption. 
Also, being confronted with the "simple solution" to a systemic problem and not give in to baser instincts is arguably part of what should define someone as a true hero - doing what's right instead of what's easy, when the two aren't mutually exclusive.    
Bruce Wayne is not responsible for the hypothetical future actions of anyone but himself.  Of course that doesn't mean that Bruce wouldn't feel a sense of personal responsibility, but that doesn't mean it's truly his to shoulder alone.

There is virtually an infinite number of possible futures that could happen and no one can be expected to account for all of them.  If Arkham or Blackgate or wherever these dangerous people go can't contain them the way they need to be contained, then the correct solution isn't to simply put a bullet it in their head, it's correcting whatever the issue of containment is; whether it's them having powers that need to be taken into account/neutralized or them being clever and finding a way to escape, you make changes to the prison or the asylum to improve it and make it more secure.  And that burden isn't just on Bruce, it's on pretty much everyone to ensure the public safety.  

A last thought on this particular remark; the Justice League animated series explored this exact idea - and I mean exact, down to where they had a version of Superman who used his heat vision to lobotomize villains, including Doomsday.  It turned out to take place on a parallel earth where everything had matched up to the conventional story and characters, until Lex became president and killed The Flash and Superman retaliated by killing him.  After that the League became the Justice Lords and totalitarian rulers of Earth.

I haven't forgotten.  I think it's one of the most important qualities about Bruce Wayne, which often gets overlooked in the live action adaptations, much to my eternal frustrations.  One skill I remember reading about, I think in a novelization of something with Batman, that I love, was his "speed reading" - which if memory serves was almost more like a meditative trance.  He'd have someone open a book and he'd stare intently at the pages; and rather than actually read the words, he'd sort of let the image of the pages and everything on them be imprinted and processed by his subconscious.
It's perhaps borders on a super-power, but I think given some of the broader things that are possible in the comics, this arguably falls towards perhaps the least implausible thing to be able to do.  And it's sort of a cool ability, I think.
Just in general, I'd love an adaptation that explores Bruce Wayne's genius more.  Because they'll drop subtle things in the comics, like how Bruce has employed magical friends/colleagues to enact layers of spells and enchantments to help conceal the Batcave to prevent people from being able to either detect it or access it through magic or other means, like teleportation.  And in another comic it was revealed that Bruce has established his own utility network for power and communication, but that it goes beyond his own personal use; where he's tied it into and in a way grafted it onto the city's own deteriorating system, bypassing certain failing parts of the city's utilities, which if they had they failed (as they would have without the support of his system) it would have seriously crippled the city.

There's a fine line between appearing intimidating and being a full fledged psycho; and too often that distinction is blurred with Batman; especially when adapted in movies.  An example of the former though was in, I think a Young Justice comic.  Tim Drake explains how he and Batman want criminals to think they have some secret location they dump their bodies in; and that they are willing to take extreme measures, so that Batman and ostensibly Robin too, have such a reputation as to frighten all but the worst of the worst of the criminal elements.  That, combined with Batman in particular being an urban legend, are the two primary conceits about the character that I think allows any degree of plausibility in Bruce Wayne being able to do what he does, without people finding out who he is and allowing Batman to be effective. 
I've never been a fan of the idea that he sees himself more as Batman.  He certainly puts on a public facade as Bruce, but I think there's still a "real" Bruce, which he doesn't often get to share, except for those closest to him. I think he could conceivably be drawn into his work to the point of exhaustion and eventually to where it will be the death of him. And it also makes me think of something else I had seen or read, where Alfred points out how, after years of being Batman, his Bruce Wayne voice has started shifting towards the lower register he uses for Batman.  But he still needs to have a grip on who he is, his values and most importantly his humanity; he can't lose sight of the line he can never afford to cross. 

That's something I didn't appreciate the first time I watched Batman Beyond; I didn't understand the motivation for Bruce wanting to retire after a particular incident in the field, where he's having a heart attack or something and the person he's up against gets the upper hand and starts knocking him around; until Bruce grabs a gun and holds it on him. Afterwards he's disgusted by how that all played out and it was a long time after I saw that, that I understood the implications of what it would mean for Bruce to feel the need to rely on something like a gun to subdue a criminal; and that it made perfect sense for that to motivate him to retire.

I don't know that I agree with a lot of that.  Most adaptation, and typically in the comics too, Barbara is kind of the outlier in the Bat-family.  She has strong ties with Batman, Alfred and all of the Robins, but is also the most independent, usually working entirely on her own.  She's not like Dick, Jason, Tim or Stephanie; she takes on the mantle of Batgirl without any real ties to Bruce Wayne/Batman, other than being aware of the existence of either.  In a lot of iterations, Bruce doesn't even know who Batgirl is for the longest time when she starts out.  I think in some adaptation, possibly starting with Adam West series, Alfred may have even learned who Batgirl was before Bruce does.  And she's typically older than say, any of the various Robins, when she starts out as Batgirl compared to when they became Robin.  Where they're usually early to mid teens, I think conventionally she's usually college age; maybe 16 or 17 at the very youngest, but I'm fairly confident in saying she's usually already an (albeit young) adult when she first starts out.  
I honestly don't think Babs sees Batman as her boss, let alone a father figure.  She has all she needs in a father and male role model in Jim Gordon; and if anything she wants to be like Jim, which in turn would motivate her to want to have the same level of association he has with Batman. 
Bruce might offer some more advanced training, but typically she does her own thing; and this really gets exemplified when she becomes Oracle and is the leader of the Birds of Prey and takes on a strong leadership role with the Bat-family, with Batman seeking her help quite regularly and her helping coordinate certain large scale, group efforts.  Robin may be Batman's protege, but Batgirl could arguably be called his lieutenant.

And I think it's a good thing for her not to be cast in the same mold as the Robins; because what's the point of having her be the same as everyone else on Batman's team?  I think there's value in the idea of her having the capacity to look at Bruce, once she gets to know him, as just a man, warts and all.  She could and presumably has called him on his BS; she would objectively know if he's going too far or losing sight of what's important. Her only aspiration with Bruce should be for him to see her as a peer, an equal.  And from that perspective, anything is possible. 
I'm not strictly suggesting they should date and fall in love and start talking about having bat-babies, though that shouldn't be necessarily off the table either.  She could conceivably have a crush on Bruce in the early days, because he's supposed to be a very hansom man.  As for Bruce "could have literally anyone," Babs is certainly anyone, she's not no one; and not for nothing, typically she's conventionally beautiful in her own right.  Not to mention she offers Bruce something that most other woman can't, which is a mutual understanding about what it is they do, the life they lead. She is one of the few people he doesn't have to be either "Public Bruce" or "Batman" with, he just can truly be himself; and the number of women he's been able to do that with are few and far between.  So I do think it's plausible for the two of them to finds themselves in the right circumstances and state of mind where they just need another person, they need physical contact and intimacy and the two have each other during a moment of vulnerability for both of them; and then afterwards they don't see a future between the two of them beyond friendship, and it ends up just being a one time thing.