Yes, you do use the counter word for only one thing. I've been looking for articles on grammar, but I can't find any that speculate about why it works; they all just say that this is the way most Asian languages work.

For example, in both Japanese and Chinese, nouns have no plural form, and in both of these languages, you have to use counter words with numbers.

I like your theory, but that system seems to use more words, rather than fewer, since you have to have the word for "small four-legged animals" as well as "knubles".