You nailed it. It was
amateur and
cute. It's obviously a place she didn't usually go to in her regular singing.
From my listening to vocal coaches, the head voice needs to be developed intentionally - it's not incidental or just gets better by itself - but so also does the chest voice (even though that is based on your speaking mode). Training one mode improves the tone and consistency of the other and gives options for moving between them and blending them.
I would suspect that Wakana (and Hikaru, though her voice was generally less developed) found themselves needing to become stronger (or more aggressive/heavier) in their chest voice singing for the sake of achieving Kajiura's vision and the need to be dominant in their parts for the (mostly non-acoustic) lives.
I do still think that they had naturally quieter/reserved singing voices, especially when you hear them sing solo acappella during interviews. As they got used to using tension for the required intensity of sound (and sometimes to keep phonating at all, especially Hikaru) the head voice also got tense and more difficult to easily/naturally sing with.
I think along the way (as you hinted) Wakana was also definitely influenced by Yuriko's voice and vibrato (since she heard it so often and called her Yuriko-sama in an interview). Possibly trying to do what she did for kajiurago sections. Notice the flutter in her vibrato. What do you think?
I watched the 10th anniversary movie again earlier this week and it was interesting to watch all the way through, especially seeing them working on their parts and coming together, and also doing the concert itself. Wakana mentioned a high-school / youth group gospel-singing experience growing up and Hikaru had some youthful singing experience as well.
Interestingly, listening to their recent interview and harmony singing, they sounded quite good and blended (of course much better than the Fate 20th anniversary
). I would propose a mostly acoustic live would be best for them so they can focus on the harmonies with a softer feel, with just a few heavier songs (maybe Magia) just because fans are waiting for those. It will also keep their voices less strained for the length of the concert (especially Hikaru).