He sees Kryptonians as a threat and there's only a max of 3 on Earth at any given time: Superman, Supergirl, and Powergirl. Superman is the strongest of the three so the most threatening but Supergirl and Powergirl are just as big of problems. You're using 90s cartoons? His comic arcs have been mainly Justice League.
And you're greatly mistaken and your information is outdated, since the only Kryptonians on earth are Clark, Kara and Jon(You know, Superman's son). Power Girl is back on earth 2.
And his obsession is with Superman, not any of the others. He only views them as ways he can get to Superman and make him suffer.
And it seems you didn't read my post very well (Look into that), since i said that Darksied becoming more of Superman villain popularized by Superman TAS and continued to the comics. Never said that it was the only example and hell, they've fought in the comics even before then.
And yes, Darksied has shown up several times in the Superman book.
Also , a lot of villains have faced the JL, that doesn't always mean they are only a JL villain.