Nobody is arguing that he is a horrible father, even with all the justifications, what I meant when I said Claude is that, like Lydia, he is a morally gray character. We don't get a lot of those in manhwa, they either super good or super bad. They portray "morally gray" as just plain old violent (the duke of the north that kills monsters, or the bloody emperor that killed his backstabbing siblings and retainers), which is dumb because the violence is usually meted only to people or things that deserve it, hardly ever to innocents or collaterals (hell, in "Please don't kill me, your majesty" her dead family ended up being a crapshow just for good measures in case you were not sympathetic enough toward the ML and in "Villains are Destined to Die" the family turned out to be under dark magic). Nobody faces the issue of not-nice characters head on like this story does (not even WMMAP). You are right that Lydia went through an immense amount of abuse and that might justify her behavior, but on the same vein, many other characters go through similar abuse and don't become warped (ie. Lyrica who Lydia forced into the exact same position her mother did her). And even after she turned back time, she is still not a moral person by any means. WHICH IS GOOD. It's interesting. Claude had the POTENTIAL for similar growth (ended up squashed but wthv) because he was not a good father from the beginning. He is introduced to us as a negative character just like Lydia, but unlike other similar bad parents trope, Claude is in the same position as Lydia because he turned back time, not the MC. It was him who could have the introspective character development and fight his own twisted nature just like Lydia, maybe not to become good (because again, he was never good) but to make up to Athy for all his crappy treatment. Anyways, all this is to say this story is amazing and I was disappointed WMMAP didn't strive for more in its character development.
I think we have a bit of a different definitions of “morally grey”. For me a morally grey person is someone who could kill, but only those who deserve it. Neither Lydia nor Claude are like that. Lydia may be like this now, but before she was purely a villain, and Claude? As I’ve said, I see him as an inherently bad person if he’s willing to mistreat/abuse a kid (even let it drown) just because he doesn’t personally care about it. He’s not morally grey, he’s just bad. Just as you said, usually what manhwas wanna show as “morally grey” is just plain violent, I think the same case is with Claude (he just can’t really be bothered to go out of his way to hurt people, that’s how I see him). As for Claude overall, I guess we just don’t see the same depth to him. To me he is exactly the same as all those “cold fathers” Korean webtoons love to write, as I’ve mentioned before, I would even go as far as to say he was a blueprint for them. Did he have potential? Most characters would have potential with a good writing. I think writers here went for cliche more than in-depth characters like with Lydia. Indeed, disappointing. I used to like him a lot when I was just starting with webtoons, probably because I like angst and art style was pretty. Now that I look back at it, he really didn’t deserve a happy ending (sorry not sorry).
Lydia was actually a horrible mother. At first she was redeemable but now looking at it, she really did awful abusive things to her daughter throughout her life. She can blame her own trauma or her first husband all she wants but she is the one who sold her own daughter. The sudden change into a "good" mother doesn't make sense when she was a drunken abusive woman her entire past life. I can't get over how awful she was.