I do think he's a little more interesting in the new episodes than he was in season 3, so hopefully they'll continue to flesh him out and improve his portrayal. Though you're right that it's a letdown after Supergirl's Lex, which was probably the most authentic live-action adaptation ever of the comics' Lex Luthor.
But that's exactly the issue -- that they chose to approach Luthor this way instead of a different way. Everything in fiction is the creators' choice, so that goes without saying. But some choices work better than others.
If by "the specific context" you mean a story where Luthor seeks revenge on Lois and her family, I consider that a bad choice, for the reason I stated above -- that superhero stories should be about the heroes dedicating themselves to helping others, not just protecting themselves and their immediate families. Plus, revenge dramas are a dime a dozen. It's a boring and petty motive. In the words of one of modern fiction's smartest villains, David Xanatos from Gargoyles, "Revenge is a sucker's game." A villain perceptive enough to realize that is more interesting than one who's driven by revenge, because revenge is a cliche.