He's broody but I wouldn't say he's annoying. I mean, until the last moment of the game he had essentially watched just about all his friends and his wife/girlfriend be brutally murdered in front of him, and then got dragged out of the hospital after having all his limbs and major body organs removed and replaced and was told to shoot people. And then depending on how you end the story he kills himself and hundreds if not thousands of other people for the sake of a philosophical question.
Actual spoilers: When Jensen finishes fighting the Hyron Project, he can choose to destroy panchaea instead of making any decision about what humanity will become. This directly kills hundreds if not thousands of workers and employees inside the structure itself, and indirectly kills anywhere from tens of thousands to hundreds of thousands depending on how cynical you are about the events which might unfold if humanity decides for itself what the best future is.
It's rare that I fall in love with a game's world that much. I'm not a huge fan of sci-fi settings in stories and games, but the gritty almost realistic version of the future that HR presented was beautiful. The themes, the stories, the world, everything. It was even all backed up by a wonderful OST. I guess I'll always be a fan of Cyber Punk over Sci-fi. I especially love the focus on human augmentation and all of the associated moral and political problems that come with it.
I fell in love with the whole series, whilst the last one has better graphic and music, the first one had a very similar ambiance, it felt awesome. Let's not talk about the second one. I love when a game brings to the table hipothetical questions like in DE HR, all those debates and confrontations, they took their time and their made a game worth playing, and worth experiencing, worth listening, worth reading, a master piece in my eyes.
How did "I never asked for this" even become a thing? It's one line of dialogue in the game that you only get once if you choose a specific option in one dialogue tree. It doesn't seem obvious enough to have become a meme.