Again, you aren't thinking about this correctly. The problem lies with GRRM, the characters aren't consistent in anything they do.
Could Arya have ever left the Starks? How could she not leave the Faceless men eventually, and how would she have done it without them putting an eternal bounty on her head.
The wounding? They needed to bring her to crisis. The stupidity? Nothing more than a little girl acting like a little girl because that's what she is. She isn't a faceless assassin, she is fallible. The healing properties of milk of the poppy? She could have slept for days, or weeks, for all we know. Time isn't consistent on the show either. Look at Littlefinger traveling from the Vale, or Brienne going all of the way down to Riverrun in a quick time, meanwhile everything else moves fast or just sits around.
Edmure said he has been in the dungeon for years. *years*. How long is that? 5 years? 10 years? We have no concept of that.
This is GRRM's ridiculous timing / story lines / prolific creation of characters and arcs.
No, the writing WAS consistent. Of course Arya left the Starks. The Starks were gone, she didn't have a choice. Robb, Rickon and Bran were dead as far as she knew, mom and pop were dead, Sansa was under control of other people, Winterfell was toast. Her entire world was gone and she had a list of people she wanted to take revenge on. So she followed the one path open to her to acquire some power, have a chance to take action on her list and maybe be somebody rather than living in hiding. Her arc made PERFECT sense until she got to the Faceless Men. Then it went all to shit.
The wounding itself isn't the issue. So she needed to be brought to crisis with her back to the wall to find who she wanted to be? Great! Showdown with Waif, injured, escapes, Waif tracks her down, Arya knows she's too injured to beat the Wait straight up and has laid a trap, Waif overconfident blunders, Arya wins. THAT makes sense. That's doable and it checks all the boxes of getting Arya to crisis where she's fighting for her life AND it keeps her in character. But to have her going to ground ready for a do-or-die fight in one episode and to have her out in public unarmed and not watching her back in the next is the dumbest thing the show has ever done. Its maybe the dumbest thing any show has ever done. It's so far out of character and so unrealistic that it would have been just as believable to have her injury caused by a crashing spaceship. It was utter nonsense.
So they're playing with moving characters unrealistically quickly while years pass for others and nothing happens elsewhere. That's just lazy bullshit, but it's forgivable and it's pretty much necessary. The timing GRRM established worked. It's what's changed between books and series that have made the time warps necessary. Some arcs are added and others deleted entirely so that things that would be happening to keep pace in various locations are not happening and causing the distortions. Big deal! I don't give a shit if Edmure is in a dungeon for 6 months or two years. If he goes in as Edmure and come out acting nothing like Edmure THAT is a problem and that's what the writers are doing now. And that is what makes anyone with a shred of common sense scream "BULLSHIT!!!!" at almost every scene.