Well, the LG paladin (according to Gygax) HAD to commit "war crimes" or he would lose his paladinhood for performing a NG act, until he atoned.
Remember that Gygax took the Elric style grimdark Order-vs.-Chaos alignment system and tried to expand it into a dual-axis system. But the single-axis origins of the system remained influential: LG was far more defined by "Law" than by "Good." There's a reason that 1977 Basic and 2008 4e had 5 alignments, and B/X, BECMI, RC, and Classic had 3 alignments, again. In the early days, the idea that LE or CG could exist was really difficult to comprehend. 4e tried to return to this idea of a line, where LG is the most lawfully and most goody of good, while CE and NG were really just the same thing (less beholden to the rules but otherwise good), LE is really just the same thing as NE (evil but not destructively so), and CE is the most destructively evil of all.
Gygax associated Order and Good with each other, even after he acknowledged that CG and LE could potentially exist. So if the Laws exist, and the Laws say Orcs must be destroyed, it must be good to destroy Orcs, right? That's the sort of extremism that happens when you go down that rabbit hole of putting Law above Goodness. But the original Paladin was definitely LAWFUL good, not lawful GOOD. Good was always in service to law, and being lawful and not good was okay, but being good and not lawful was blasphemous.
Thank HEIRONEOUS that's no longer the case, and alignment is not a straight jacket. We can tell stories of Daredevil-ike Paladins that skirt the line of their faith and wrestle with whether their acts are truly good, whether they go far enough, or whether they've gone a step too far.
Honestly, in my mind, if you want to play a Paladin with atonement rules and internal struggles of what to do when, take a watch of Marvel's Daredevil show (previously on Netflix, now on D+). Given that Matt Murdock is a LAWYER by day and a non-killing-but-quite-violent vigilante by night (not to mention one with Blind Fighting, because Justice is Blind), who goes to church to confess his sins to his priest because he struggles with whether he's doing the right thing or not by beating up criminals and taking the law into his own hands - I'd say he's the perfect example of a LG Paladin.