I think a few things may factor in come into factor.
I always thought it was because of the harsh ultimatum Illumi was placing with a cocky sort of nonchalance about either him or Alluka dying due to Killua’s actions wasn’t. If any family member died because of Killua’s involvement, it’s not unlike him to blame himself and harbour it for a long time, Illumi was correct in saying that.
I’d say Killua still loves his family despite all they’ve done, and I doubt he’d want to see any of them dead. While he was willing to sell his brother out to Morel and the association, and at the start of hxh when he wanted to capture them for their bounties, he’s never expressed any desire for any of them to die, not even in jest.
So that mixed with frustration, anger, and just everything that’s happened so far probably got to him. Milluki did say “In the end he’s just a crybaby, mentally weak.” While I wouldn’t quite say he’s mentally weak, Killua does feel very deeply and often loses control of his emotions. Before he would just get angry and kill people for the release, but now it often comes in tears.