G. C. Rota has analysis of justifications of the univariate normal distribution in his Fubini lectures, problem 7. DOI https://doi.org/10.1007/978-88-470-2107-5

G. C. Rota has analysis of justifications of the univariate normal distribution in his Fubini lectures, problem 7. DOI https://doi.org/10.1007/978-88-470-2107-5_5 https://doi.org/10.1007/978-88-470-2107-5_5, a great place to sense borderline topics in the field.