It's because people simply don't know how to play Sona. She has a lot of stuff to her and to actually be able to carry/win in mid lane you need to know everything to her. Easily one of the most versatile champion in the game with her passive.
Most professional players probably don't know that every time Sona Q's, her and everyone around her has extra damage to their next auto attack.
After Sona gets Lich Bane she hits a major power spike similar to Irelia and Sheen or just Trinity Force in general.
But yeah, AP Sona isn't outdamaged by Morgana/Taric/Leona and I've played enough games with Sona (and even against Morganas in mid lane) to know that.
At low levels it's easy to kill Sona because she has almost as much health as a minion though she does have sustain and if played right it's not hard to get kills at about level 3 - 4. I've personally obtained kills at level 1 against people such as Morgana, Vel'Koz, Leblanc, Akali, Katarina, and Yasuo.
Though I'm sure another reason why AP Sona, once you know how to play her is so effective is because people typically have no clue to counter a girl with a floating etwahl (or harp) that just pokes you down until you recall lol.
tl;dr Sona has a really high damage output but to be effective with her you need to master her. I guess it's sort of like Kassadin? I haven't played him much but it wasn't the easiest Akali--mode champion in the game and since I believe I played around 120 games as Sona Support before even trying mid lane it was easier. (besides one time I had to switch lanes and go against a Silver ranked Lee Sin while I was still level 15)