One thing I was ruminating on as I rewatched it that Klaus still isn’t someone the show insists we root for, and yet his characterization makes so much more sense here that he becomes more sympathetic.

I wholeheartedly agree with this. Klaus is still the bad guy, still obsessed with power and willing to betray and manipulate everyone. Yet because Elijah and Rebekah love him, the audience loves him. It's a joy to watch.

I loved Klaus even on TVD, but I agree that he couldn't go on being the eternal adversary. He basically ran out of story after Bonnie dessciated him at the end of Season 3. It was either death or spin-off. I'm so glad we got spin-off. :)