Well firstly, Tomb Raider got the exact same criticism. Secondly, I haven't played Quantum Break so I personally can't comment. What I've seen of it though is that it goes for a much darker, serious tone than Uncharted. Like you said, it takes itself seriously, and whilst UC definitely takes itself seriously there are comedic and light-hearted elements.
Personally I can look past Drake's killings because the character is so well done, and it's a bi-product of its TPS and Indiana Jones roots. Like I've said before, I don't agree with Jim's opinion, but he holds it because he personally cannot look over the fact that Drake kills so many people in the game/s.
Drake doesn't do any of that. The player does. The player is the one who kills.
Jim just can't come to terms with his own inner monster and instead blames Nathan Drake, a puppet who uses humour to combat the dark deeds Jim makes him do. Nate finds solace in his close relationships and they help him overcome this hellish existence. He thanks Straley for setting a line in the sand that Jim cannot cross, though he has tried many times to murder Nate's family.
Jim would have us believe in his opinion because it absolves him of any wrongdoing. But we know better.
We know Jim is the true monster here.