Braavos is home to a death cult and rich merchants, so it wouldn’t surprise me if bystanders just see her as either a wounded thief, or a fated victim of the Faceless Men.
I don’t think he’s going through a lot of trouble.
Something happened with Ayra where he owed her the lives of a few people. She then blackmailed him to help train her or she’d order him to kill himself. He would have had to do it due to his beliefs or some shit.
He only truly let her fuck up once and then punished her for it. This time, she fucked up again and the waif asked him for permission to kill her. He implied yes.
Why Arya? Because she’s actually a good recruit for it.
She’s young, has physical aptitude, and is mostly comfortable with killing people.
She’s got warg blood, and while magic goes explicitly unexplained in GoT there’s very much a magic-begets-magic thing going on, so she’s probably a good fit for Faceless Man hoodoo.
Maybe most importantly, she managed to casually befriend a Faceless Man even after learning what he is. That probably almost never happens in Braavos between the fact that they normally don’t stick around after completing their job, and the fact that the locals are all in awe or terrified of them.
The Arya storyline is one of the deviations from the book as far as the whole flavor goes. In the book, there’s no Jaqen, no blatantly antagonistic Waif. She hits a lot of the same beats, but it goes deeper and it goes longer, and it hints at her ‘cheating’ part of the training in a way that might or might not be exactly what the FM wanted out of her.
There’s also a Faceless Man somewhere else in the story, who people theorize might be Jaqen, but saying where might spoil a TV plot point they didn’t get to yet. The flow of time in Thrones is convoluted.
Yeah next episode you see Arya jumping from the ledge with different clothes on & no signs of being shanked…So my guess the one who got stabbed is Jaqen, he decides to give his life for Arya…I think Jaqen is possibly Syrio the dancing swordsman. They developed a nice teacher student relationship & Syrio witnessing what happened to the Starks decides to train Aria to become the perfect assassin to get her revenge.
Other theory is Jaqen is not Syrio but is still obligated to Arya because she save him in Season 2.