I've actually been playing through the Twinshot arc for the first time ever recently, and even without having years of animosity built up between me and her there's some definite major issues with it.
The first thing, as has been mentioned before, is that it's a tutorial that's about five levels too late. All the "go talk to Ms. Liberty" and that nonsense is totally worthless once she introduces herself, since obviously we've already done that. Someone somewhere mentioned that she oughta introduce you to BAB, which would make more sense in terms of design and aesthetics; I never see anyone at BAB so maybe people don't even know he's there, and Twinshot herself is a low-level "street" type hero, so her knowing (and showing off that she knows) BAB seems more probable. However, the tutorial aspects of those early missions are also just half-baked. One bit of a mission has you go to a police station where, once you get one in your contact list, you can interact with a detective and get the police scanner. The mission DOESN'T HAVE YOU DO THIS. Another one has you go over to Wentworth's and another to AE, but neither of those in any way explain what these locations are; lots of people will probably check it out themselves later on, but why have this in a mission that has nothing to do with these things? I'm on this mission, I'm not gonna stop now and run some player content in AE.
The other thing that annoys me is that the enemies in some areas all con higher threat and as bosses (but they clearly aren't actually) and you often fight only one or two at a time. Playing through them at the appropriate levels, they are slightly harder to kill than normal enemies but don't seem any harder to hit and don't deal appreciably more damage. Maybe I just got lucky, but it seems to me that they made all the enemies harder at face value to make your character look more powerful. If Manticore was really testing you, it didn't feel like much of a test. Why not have gangs of dudes and some ACTUALLY higher tier enemies like every other mission I've done? Hell, the packs out on the streets follow this better.
Some of the mission design and dialog is bad. The one mission at like level 10 I think where you go to the new base was a standout because of the almost-deliberately-misleading lines about what's going on. You go there for another wasteful tutorial trek, but then it's actually a mission. Why can't Proton tell me where his relays are that he made? Why does he tell me the Skulls leader is trying to escape when the guy is just hanging out on the top floor? As far as dialog, I'm on the side of the fence that doesn't like when my character's lines press too far into telling me what kind of person I am or what I think. This mission arc isn't the worst of that, but it gets out into that field.
All that being said, the characters only somewhat annoy me. Twinshot herself comes off as a jackass but I also get the impression that it's her trying to act tougher and more "world-weary" than she really is. Grym is fine if a rather tired, archetypical personality; Proton I don't like because of one throwaway line where he's like the big hero "of his own timeline," but here he's not particularly powerful...why exactly? Flambeaux and Dillo are both the most grating of the group since neither are funny, but I think Flambeaux was done deliberately. Somebody just went too overboard on the "goofy alien" with Dillo.
Now, at the end of this, the story arc does have something that I actually like. It puts my character right in there with other characters in the game world and puts a story to it that actually has my guy be somewhat important for the outcome. I show up and do the stuff that actually leads to the ending of each of these missions and, although I'm being told to do these things by other characters, the mission flavor text actually acknowledges that it was me that did it and characters are suitably impressed and appreciative.