I have always loved the idea of ANBU!Sakura (in canon, that is) for much the same reasons that you’ve listed above, but also as a sort of… Hokage training? Or leadership training, rather (because, while my heart of hearts will always love the idea of Hokage!Sakura, there’s no way that she’d take the role from Naruto until he willingly gives it up–realizing that it’s not what will make him happy–or if he is unable to).
In DoS though, her career trajectory isn’t the same, especially since her starting point is different, so I’m not sure if ANBU would be the best/desirable path for her. Or at least not anytime soon. (Which is not to say that she wouldn’t be nominated for ANBU, but I don’t think she would accept it in the near future)
Her overall goal in life–as stated in DoS–is to prove herself as a capable kunoichi, such that she will one day have a symbol on the back of her clothes much the same way clan kids carry their clan symbols. Hers is the empty circle waiting to be filled. In canon it’s a little different–her goal is more… emotional, I guess, in that she wants to be a capable kunoichi to prove that she can stand alongside her two teammates.
Since she wasn’t put on a jounin led team in DoS–more specifically, the Team Seven that would, on their first C-rank, free a small country from the grips of an evil despot–and instead started in the Medic Corps, her main obstacle is that, to put it crudely, name brand recognition.
In DoS she starts as a medic and trains toward field readiness, whereas in canon she started as a field nin and directed her path towards medicine. She’s at least a year or more behind on field experience and spreading her name as a kunoichi who can, frankly, survive a lot of weird shit–Shikako got that instead (compounded, seeing as how I’m pretty sure Sakura never literally died at any point). As of Chapter 137, which is pre-Mist Chuunin Exams, DoS!Sakura hasn’t had any international exposure that canon!Sakura got by being on Team Seven.
She has to build her reputation more slowly, but because of that she can do so steadily and more carefully.
ANBU kind of defeats that purpose. Don’t get me wrong. It would–even just the training/audition–greatly aid her in getting field capable (ninja boot camp, level up in just two weeks!) but it would all be anonymous.
And how would she explain her sudden jump in ability? Either that or she’d have to already be capable enough that the ANBU training isn’t out of place, and by that point she’d probably have enough field missions under her belt (as either Tsunade’s second apprentice or as part of Team Anko?) that she wouldn’t need it as much.
Sasuke and Shikako want the anonymity: they’re at a point where their reputation exceeds their skills (or, rather, the risk of having such large reputations exceed their skills) so ANBU lets them use said skills without being hindered by their faces and names.
Sakura isn’t at that point yet. And given she’s not technically a Lucky Seven, that may never happen. Her reputation will probably match her ability throughout her career without any weird imbalances or spikes.
Of course, there are other reasons why people join ANBU, but I don’t really think they much apply to Sakura in either iteration.
As for the super-strength technique, if she were to join ANBU, she’d train herself against using it while wearing the mask (the same way Shikako trained herself against using shadow jutsu and other, easily recognizable techniques). I’d say her pink hair would be the most obvious thing, but I suppose in ~anime world~ pink isn’t explicitly weird? Harder to hide than regular brown and switching between braid and bun, but hand wave that.
I hope I’m not raining on your parade, anon. As I’ve said before, ANBU!Sakura is a wonderful image and I can only ever dream of being a fraction as badass.