And honestly, when you're terrifying enough to install that kind of fear in the kind of people Batman deals with, you can't expect him to not be equally or even more terrifying to ordinary citizens.
But that's not the in-story logic behind Batman. The core idea is that Batman presents himself in a way that's only terrifying to those with guilty consciences, to those who fear retribution for their own wrongdoing. Also, that "criminals are a superstitious, cowardly lot," that they're dumb enough to believe Batman might be a demonic figure out to punish them for their evils, while law-abiding people are likely to be better-educated and not as prone to superstitious dread.
The Nolan films got this right. Batman's first big action scene was played from the perspective of the mobsters, and Batman was treated as a horror movie monster, a vaguely glimpsed figure striking out of nowhere. He was terrifying to the mobsters because he was
coming after them in a way they couldn't see or defend against. But the good people of Gotham saw him as a symbol of hope, someone who would protect them from the people who preyed upon them.
Also, as you have pointed out yourself in the past.... In the comics there have been different versions of Batman, or any hero for that matter. There is no defined 'point' to Batman, because there have been several takes on the character. You prefer the version that isn't like what we saw in BvS. And that's all good.
It's not that random. There are some elements you can change without losing the essence of the character, but there are other elements that are key to that essence and need to be preserved in order for it to be a valid interpretation of the character. You can redesign the chassis or the seats or the dashboard of a car and it'll still be a car, but if you replace the engine with a kiddie pool or the wheels with wooden crates, then it's not gonna work. I mean, Sherlock Holmes in present-day New York is still Sherlock Holmes, but a dimwitted Victorian gentleman in a deerstalker hat is not Sherlock Holmes. Some changes are variations on a theme; others are just missing the point.
This was Batman's character arc in the movie--he started out in a really dark place. By the end of the movie and in Justice League he was the Batman who also inspires and gives hope. (I'm sure some people will argue how well this was executed, but it was certainly the intention.)
Okay, I'll take your word for that. I found the execution staggeringly incompetent, but if the intent was that that wasn't the right way for Batman to be, then I'll grant that it wasn't missing the point.