The definition under "A Brief Recap" seems incorrect. The sample size doesn't approach infinity, the number of samples does. I'm in a similar situation to the author, I skipped stats, so I could be wrong. Overall great article though.

It is correct in the article. As the sample size approaches infinity, the distribution of the sample means approaches normal.

https://en.wikipedia.org/wiki/Central_limit_theorem

Yes indeed, if the sample size approached infinity (and not the number of samples), you would essentially just be calculating the mean of the original distribution.