Japanese might treat barley as a wheat, but in English and in biologic taxonomy they are different. Wheats are in the genus Triticum; barley is Hordeum vulgare. So I'm guessing that the word translated as “wheat” should be translated here as “grain”.
Yup, you're exactly right here. In the original text, the phrase "ムギ"(mugi) is used, which while collectively can refer to many different types of grain, it generally is used to refer to Wheat and Barley. Wheat in particular is known as "コムギ"(komugi), which incidentally is the name of one of our protagonists. Barley is known as "オオムギ" (Ōmugi). We felt that grain is too broad here as it was specifically refering to grains with "ムギ"(mugi) in the name, rather than other grains such as Rye or Oats, so we chose to translate it as Wheat as it was already brought up as an ingredient in beer when Weizen was introduced.
TLDR, specifics of language hard and we made a choice at the expense of scientific accuracy