Your leadership point doesn't carry weight right now, not only has he been poor, there's a underlying toxic, Ronaldo-esque thing going on where he's acting up when decisions are made that benefit the team over him. He's always had that, and it's why both Klopp and Slot have struggled to limit his minutes when it's been obviously necessary.
Players look up to him and I'm sure he's a great example as a model pro for looking after himself and extending his shelf life, but let's not pretend he has the same level of humility and self-awarenesss as the likes of Milner or Virgil. If he looks pissed off it's not because we're losing first and foremost, it's because he's not performing, and if he's hooked or dropped you can guarantee he'll be annoyed because it's inhibiting his personal goals as much as anything. That might be unfair and the double edged sword of top class players, but he's far from the captain you're aimlessly selling him as.