Ah~haah, that's what this is about.
Miu has a
pre-disposition towards satsujinken. This is why the Elder took her on that training journey when she was really young - to try and 'train' katsujinken's mentality into her.
As for her going off with Shou, he had already beaten Takeda, Freya, and others without a second thought and Miu 'snapped'. She was honest-to-goodness, trying to kill Shou for what he had done to her friends. Miu cares a lot about people - doesn't matter who it is - it's the reason why it's a conscious effort in Katsujinken to NOT kill someone even in a heated battle.
Her surprise in Kenichi's victory over Shou was shared by nearly everyone on that island. Shou was the defininte odds-on fave to win that one. Kenichi was the underdog. Even though one may root for the underdog to win, that doesn't nessecarily mean that everyone
believes the underdog will win.
As for the masters of either side being attracted to either Kenichi and/or Miu, they are all martial artists - they can recognize talent and hard work. Hell, Asgard was all compliments and smiles on Kenichi even when he was fighting against Apachai. As for why the Satsujinken people focus on Miu, they can tell that she has to consciously hold herself back while Kenichi doesn't really have that problem. And when I mean 'to hold back', I mean to hold back an attack's power because her targets are in lethal zones. Kenichi subconsciously doesn't target a lethal zone, however, Miu does. And the Master classes can obviously tell.
This is the entire debate with the manga and believe it or not, in some martial art circles - even if Martial Arts were originally intended to maim and kill an opponent, does that mean that they have to? If you are already so strong and you know the smuck with the gun trying to hold you up will die from one of your blows to the diaphram, what do you do? Do you kill him and be on your way, or do you disarm him and be on your way?