The myth of 30-samples goes back 100 years to a misreading of a 1908 paper by Gosset.
https://charliekufs.medium.com/30-samples-standard-suggestion-or-superstition-ee5e78fb6e8b
Since then, the suggestion has been justified in a variety of ways, as you describe. It is too commonly taught in college courses for non-statisticians.
