I appreciate how earnest the game is with its corniness. Not just the writing either, but Overwatch as a whole. Character designs, map locations, voicework, sound design, music etc.
I mean, there are some "darker/deeper" elements to the story/lore, but I think they're wisely downplayed. The impetus is put on the player/fan to learn more if they really want to.
There's a charm and likeability to the whole thing, which is not something I can say for a lot of games - particularly shooters. I never expected to care for the characters to the extent that I do. I do find some lines amusing as well.
On the subject of Omnics, I do think we're due an evil/corrupted one. Without in-game context to the Omnic Crisis, Torjborn, Zarya and Junkrat's attitude to Omnics comes off as a tad extremist - particularly if you base Omnics off Zenyatta and Bastion.