Alleria did not follow her mother’s footsteps into becoming Ranger-General by choice: when Lireesa still lived and tried to pass on the mantle to her, Alleria refused, claiming she would be better and more useful in the field rather than commanding; and a second time when her mother died and she, being the eldest Windrunner, was expected to assume the position. Alleria never wanted to be Ranger-General, even though she was expected (and raised) to. She did not want the burden of leading, neither did she see her strongest skills as being in leadership, for she believed to be of more use serving than leading. But if she did not inherit her mother’s position, Alleria was the one to inherit her bow nonetheless.
Thas’dorah, “Valor of the forest”, was the proud heirloom of the Windrunner family, carved from the mother tree of Eversong Forest. It was handed down to the eldest of each generation of Windrunners, and it is a legendary bow, said to be able to turn even a mediocre archer into a master marksman and make a master truly peerless. I believe the later to have applied to Alleria: she already had skill almost unrivaled, regardless of the bow she would be using, but when she wielded Thas’dorah, she was without match.
But to Alleria, wielding it went well beyond just being unrivaled as an archer. Carrying the legacy of her family was very important to her, and Thas’dorah was almost this legacy made physical. It was daunting to wield it, in the same way that carrying the very name Windrunner was daunting at times; but it was also something she did proudly, with so much confidence and skill that none would ever think it truly weighted on her.
More than that, it belonged to her mother before it was passed on to her, which would make her bond to it all the more personal. Canon seems contradicting as far as Lireesa’s death goes, but in the end it doesn’t matter. With her mother passing and the bow falling to her, Thas’dorah would mean even more to Alleria, and her desire to wield it well and honor its legacy would only be greater.
The bow would always be with her, and she’d take good care of it, so that it would not only be kept in good state but utmost pristine, fitting for the legendary weapon it is. Alleria probably spent hours polishing it, making certain the string was in good condition and well attached and replacing it periodically, storing it safely and with all care in the world when she would not be wielding it.
She’d never part from it if she had a choice. Even when escaping her prison with Locus-Walker, Alleria’s mind goes towards searching for her bow before leaving Niskara, but she could not, because they wouldn’t get a second chance to escape the Legion, so she has to leave it behind or she wouldn’t leave at all.
Knowing the bow was later retrieved from the prison gives her mixed feelings. Yes, she’s glad it is no longer in the demons’ possession, but at the same time neither is it with her anymore, and Alleria feels rather possessive about it. It is her bow and a family heirloom, so it is understandable why she would be — it isn’t about needing it, but about the emotional value it has to her.
She would want it back, no matter who wields it. And while she would accept others having it because she must, it wouldn’t be anyone else, and she’d surely be against orcs and trolls having it, specially if her mother died in the Second War. The least displeased she’d be would be is if the one to have it was a sin’dorei.
Even then, she misses her bow. Yes, she has another and yes, she’s skilled enough that few can match her regardless of which bow she’s using, and yes she doesn’t even need a bow anymore because she can use her void powers, but Alleria misses it nonetheless. Even if she can use the Void, she’s still a ranger at heart; and if she has a different bow now, it isn’t Thas’dorah, which she received from her mother and which meant much for her family, and for her personally.
Post Legion, Alleria will get it back. Like I said, it doesn’t matter who has it: it’s a family heirloom, it belonged to her mother, it’s hers. If the wielder doesn’t give it up voluntarily, she will seek it to have it returned; and she will fight for it, too, because this bow means a lot to her, so she will have it back even if the one who has it would rather keep it. Gameplay being different from lore, I think it’s also valid to state that was she to meet a Troll or Orc hunter, she would not let them have it, not even during Legion.
Post Legion / throughout BFA / from then on, the bow Alleria wields is her bow, Thas’Dorah, though how she got it back can be discussed accordingly if relevant.