Why do the tropes keep insisting that Shiro didn't have multiple personalities?
Maybe I missed something and so I won't go and edit everything (also, I'm very, very lazy), but despite her statements near the end, we repeatedly saw Innocent Shiro's thoughts during Ganta's stay in Deadman Wonderland and they seem to be far different from how the Wretched Egg behaved. Even assuming that it wasn't an inaccurate/incomplete translation or Original Shirou/Wretched Egg lying to Ganta to make him more willing to kill her, we still saw her work on her Branch of Sin while alone (IIRC), which she wouldn't need to do if she had the memories of the Wretched Egg, and we even saw multiple Shiros in her very mindscape, one of whom was confused by what was happening. Heck, we even saw her Wretched Egg personality appearing during her initial testing period, apparently to take all the pain that Original Shiro couldn't handle. I guess its possible that Innocent Shiro still has a subconscious link to her other personalities, which would explain why she knew Ganta was there, assuming that the Director didn't tell her for the sake of seeing what she'd do, but that Shiro didn't seem to be the Wretched Egg unless something happened to the Mother Goose system, at which point she pretty much instantly shifted over (as Yoh found out when he accidentally got a bunch of guards slaughtered by Wretched Egg Shiro when he tricked them into shutting off the power in that area). Barring this being a massive retcon or deliberate misinformation from a Death Seeker, it seems more likely that Shiro managed to reintegrate her personalities as she started forcing her way past Mother Goose's lullaby.
The idea that Shiro did not have a split personality seems to stem from her one comment that "there is only one Shiro". This comment could be interpreted to mean that there has always been only one Shiro. However, given the above arguments for why she likely did have a split personality, it seems more likely that she meant that "now, at this point in time, there is only one Shiro", due to the personalities eventually merging together. Admittedly, the manga probably could have done a better job of clarifying this.
Why is the Mother Goose system a problem for Hagire?
Yes, the key is tricky, but we already saw that shutting off the power at least turns off the local Chorus Boxes, and flat out destroying the core worked perfectly well once Shiro got access to it. Is there any reason why just blasting at the main node with technology or Branches of Sin wouldn't work fine? Heck, coat it in butter and call in Masu, if you want a joke ending.
How was Ganta convicted in the first place? Yeah, they showed doctored testimony about him bragging, but the classroom was trashed and their was blood everywhere. It would take something like a bomb to do that. How did they figure Ganta killed everyone yet was unharmed himself. Even assuming a corrupt judicial system that sounds like a stretch.
Sadly, it's not. While it might be an exaggeration, if the entire system is corrupted, the entire thing is just one big kangaroo court. With no real family, they could frame him for nearly anything and then fabricate any evidence they wanted to for the public to make him look guilty.