I wasn't sure myself until I translated the postscript. I see this as a pushback against the overt sexualization of young characters in manga. Aside from her physical maturity, there's nothing sexual about Aoi-chan. She's mentally exactly as old as she is, and how she sees her would fits completely within that maturity level. She has a childlike, idealized, and cliched (note all the shoujo manga tropes) view of what romance is, and it's closer to yearning to become adult/admiration for adulthood than romantic love. In the postscript, the author uses the word 憧れ (akogare) which is basically wanting something and admiring it, meaning that Aoi-chan wants to grow up, not necessarily by doing anything adult-like, but just an unspecified, undefined yearning for it. Not much different than wanting to be a mahou shoujo or a masked hero.
I think we as readers are conditioned to expect certain traits from the characters we see, and at least my initial discomfort came from that. Because I brought into this story those baggages, I saw things that were not intended. Reading it more closely (by translating it line by line) made that clear to me. So this title requires a little more investment than some others, I think.
In the MD title page, I was very adamant that this isn't a romance, and I still stand by that. There's certainly no romance from Natsumi, and any such feeling that Aoi-chan might feel is simply an infatuation toward something she doesn't have and doesn't understand, yet.
Sometimes, you have separate what is there and what you've brought in, and see the story for what it is.