He didn't shoot thugs with machine-guns for starters, and while it was reckless driving nobody died by the tumbler (cf Alfred's line "it's a miracle no-one got killed"). Dent and Ra's are riding on the line, but the monastery stunt must have roasted some ninjas, even if technically he wasn't batman yet.
What people complain about is Batman, in the suit and with 20 years of career behind him, mowing down vehicles with .50 cals. Which was a bit strange indeed. I did notice that he never starts the shooting actually, but he reciprocates swiftly when shot at. Except for the Russian, but that was part of the DKR homages.
Whilst not the same for everyone, I think the general gist with these things goes a bit like this.
The film is widely received as good by the public, then things that might normally **** off a "normal" comics fan would be glossed over - they wouldn't impact the film experience so much. And I would also wager that the reach and tenacity of general opinion, especially negative opinion (whether accurate or not) is far, far wider and more prolific now, then it was 10 years ago (Batman Begins 2005 I think?)
However, since DoJ was so divisive, with a widely panned reception from the general public, things like Batman "breaking his rule" (despite the fact that it hasn't actually been established in the DCEU if he even has that rule) stick out like a sore thumb. If there had been a more straightforward story, less risk in the film, a different tone where humour is the main concept surrounded by less mythology etc then perhaps it would have been more overlooked than it has been. As it is, it is just one of many (inaccurate in cases) complaints and is therefore used as a staple argument. The wider reach of the general opinion is so immediate and gospel like nowadays, that the "general opinion" is becoming worth less and less, I think. People are so very ready to put things in two distinct, extreme categories. Apparently there is no middle ground.
Anyhow, I watched Begins the other day. Still a phenomenal film, I recognised even more influences from the comics than I did back then, but I noticed some very weird things, like the killing that you mentioned. I knew it was there before, but it was so...blatant this time.
The setup of Batmans gear seemed so shoehorned aswell?? Like the first case he opens contains a grappling gun. The second case has the armour almost fully formed. The reasons they give with exposition sort of fit but doesn't quite feel like it fits. The fact he chooses to dress as a Bat doesn't quite gel right within the story I don't think, even though on paper, it makes sense in the films context. Tiny little things that just started becoming more apparent. The movement in the suit seemed...weird. Especially compared to how Batman now moves in the DCEU. Totally different.
Still love those films though. The "grounded" take on the Batman was exactly what we needed at that point in time, which is probably why it resonates so well with people.