There might be a gender issue (or past or more industry-wide gender issues) at play here.
You are using the pronoun "she" to refer to this person, which leads me to assume she is a woman (you might be using it to make the situation less identifiable, in that case ignore this answer). Furthermore, you have a username typically associated with men (again, if that's not the case, ignore this answer).
There are many challenges that women face in society, and specifically in the software industry, that us men are typically not aware of. We like to think our industry is super inclusive because we know for sure that a man sexually assaulting a woman is not tolerable in the software world (hopefully this is the case in your company). But there are plenty more issues, less visible perhaps but that still affect women. One of them is their peers respecting them less or not perceiving them as with the same level of knowledge or authority just because they are women.
Behaviors you've mentioned, a potential explanation, and some very imperfect actions that you might want to talk through with her before implementing:
- She had some concerns with the direction of one of our projects. Rather than directly state what those are, she said "it may be better to" followed by her suggestion.
It sounds like you respect her technical opinion, and would very much like to hear her thoughts. However, you can't assume that just because you acknowledge her technical expertise every other man will do so. She probably encountered multiple situations in the past where her concerns were disregarded and her expertise ignored, simply because she's a woman.
On one hand, you can empower her. If you refer to her for technical expertise, you are establishing to her male colleagues that her technical expertise is valuable. Imperfect because she deserves her opinion to be valuable because she's technically good, not because a man says so.
On the other hand, you can work with your company (HR I guess) to figure out how to control this behavior. Imperfect because they typically don't have a big motivation to fix this.
- Not willing to speak up in scrum meetings or PI planning sessions unless probed by me or another more senior team member.
Same as above, there's a high chance she was ignored in the past when she spoke up.
Don't put her in a position where she has to speak up every time. Offer other, alternative ways for her to voice her opinion. Imperfect because she shouldn't need to be protected from others.
- When needing something from another team, always asks me first and copies me on emails for requests I feel she can handle herself perfectly.
If she's asking, she's probably been talked to in the past for being too assertive, or is afraid of being perceived as too assertive, since this tends to hurt male egos. Or perhaps she's trying to borrow your authority (as a man or more likely as a manager).
You could tell her that she doesn't need your permission, but if she ever feels she needs to borrow your managerial authority (be sure to make this about you being her manager) she's entirely free to cc you into any email, and talk to you whenever she needs. Protect her and empower her, as any good manager should do for any report. Imperfect because she shouldn't need more than her male colleagues.
Honestly, men discussing women's issues like we had any authority on the subject is actually part of the problem. But I'm writing this answer because it's probably something you need to hear, and I'd rather you hear it from a man than don't hear it at all.