Yes, that's right. なる can be used with adverbs e.g. 詳しくなる. 大きくなる
And also it can be used with nouns in the form of noun+に+なる , e.g. 先生になる, same pattern in English.
Yeah, it's different from English in term of adverb vs adjective, but In fact, I guess we can also ask "why in English we don't use adverbs after 'become'", e.g. He became happily. instead , we just say : "He began to live happily". :)
Hope it helps