Q:

Why does this imply normally distributed errors?

Why does the following process imply that the errors are normally distributed?

A fellow researcher fits a simple linear regression model $Y_i = α + βX_i + U_i$ by least squares, but has
  doubts that the standard assumption $E(U_i) = 0$ is correct for her data set. She decides to try 
  test for this assumption by calculating the residuals from her regression $\hat{e}_i = Y_i - \hat{Y}_i$, and then
  performing the usual $t$-test of the null hypothesis that their (population) mean is zero,

A:

The assumption of normality comes when applying the t-test - that assumes normality (though its type I error rate is not especially sensitive to normality).
Least squares is ML (and efficient) when you have iid normal errors but you can calculate least squares estimates without normality (they're still BLUE in the case of non-normality -- though that's not necessarily a particularly useful property if the distribution is so far from normal that all linear estimators of the coefficients are poor)
It's when you are deriving things like tests, confidence intervals and prediction intervals that you tend to make a distributional assumption, and you do make a normality assumption in deriving the t-test.
[It's not necessary to assume normality even then, but without assuming normality you will end up with some other test than the t-test, even if you use the same test statistic. Note further than normality isn't the only assumption involved in this.]

