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.
Not sure what you're stating but it seems to be made of straw.
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.