This is just my opinion, but personally I don’t like labeling Chihaya as a kuudere. I feel Chihaya’s personality as a whole is more complicated than to describe it simply by a character archtype.
Of course, it depends how you see a kuudere personality.
Some examples of kuuderes I use are Rei Ayanami from Evangelion and Kazusa Touma from White Album 2. Both are not very open about themselves like all kuuderes. However, the examples I use are also pretty harsh/cynical towards the protagonist.
Chihaya’s not that harsh/cynical towards the protagonist. Neither is she emotionless or completely detached. During the beginning of the games, she’s not very open about herself (given her past) and just wants to have a professional relationship with the producer since she wants to focus on singing (it’s one of the few things she cares about and to distance herself from her past). But, overall she’s a nice person and says please and thank you.
She’s sometimes a bit direct (outside of past or romance) by asking/telling you whatever is on her mind or tell you what she wants or doesn’t like. But, it’s never really done in a rude or offensive manner. It’s just the way she is. She likes to express herself to you because she feels comfortable being with you.
The few times you get scolded is because she cares about you, but all lovers do this anyways. Of course, this happens in later parts of the game. There is one case early in the game where she’s disappointed in you for lazing around during her day-off (while she’s busy), which the producer apologizes and says she’s right because I guess he’s disrespecting her by lazing around while she’s busy.
But, overall she’s a nice person that’s not very open and too focused on singing to see things in a different perspective. Of course, over time she open herself towards you, cares about you more, and opens up her perspective much more than just singing.
