As a point of history, either a US Navy Corpman or Navy Seal executed a young prisoner of the battlefield of Afghanistan. Large swaths of the US population, including some notable leaders, had no moral qualms, about this.
Seeing as the overriding population of most Commoners is 'Neutral' I'm not shocked.
Evil people would almost unanimously support it, as would a large number of neutral people (with some being undecided, or seeing both sides of the argument, and some being repulsed. Good people would be almost unanimously opposed.
Except, in their view they are not LE.
For sure. Very few evil people think they're evil. They're almost always working towards a 'greater good' or rationalising their actions away somehow. Very few morally repugnant people actually see themselves as morally repugnant.
To be fair Frank Castle is under no illusions that he is evil. He's literally met the Devil in one comic and was totally unsurprised to hear that he's on a one way ticket to Hell when he dies.
Beside..
.....Outside, Once Upon a Time in the West, Charles Bronson characters strike me
as Chaotic Evil.

.
No regard for the law, never forswears vengeance, and is often sadistic.
That depends. Do they follow a code? Lawful people have respect for family, tradition and honor. They follow a code that (to them) places them above 'ordinary' villains.
The Punisher follows just such a code. He thinks that as long as he only murders, tortures and so forth criminals, he's different from them. He values honor, family and tradition (his primary motivation is the murder of his own family). He doesnt kill those he sees as innocent, but criminals on the other hand get no mercy, pity or remorse. He's intensely driven, predictable, and focussed.
Compare to say.. Anakin Skywalker (Jedi, not Sith!) who is CG. Unreliable, unpredictable, headstrong, hard to control, individualist, reckless, but with a strong moral code. He doesnt go around killing and torturing people (yet!).