Franke Potente being offed at the start of the second Bourne film always rankled, and I could never enjoy the films after that, because they'd lost any semblence of humanity, they were just about a blank faced killer wandering around killing other blank faced killers.

That's often the only thing that could credibly give the character the motivation to do so - in which case there's nothing "arbitrary" about it as an artistic decision.I dislike anything where characters are just arbitraility killed in order to get the hero back in the game.
In-universe, well, arbitrary things happen in real life. Sounds as though you might want to simply avoid action/adventure sequels where the first had happy endings.