Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model , very similar to the textbook model discussed previously.  The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies.  However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations.   In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence . On the contrary, the observations are dependent !
Would it ever be the case that the significance tests of the regression coefficients would come out non-significant when the overall F-test did come out significant? What if, for example, you had a factor with three levels, A, B, and C, with means 3, 5, and 4. If C is the reference level, could it be the case in the regression model that neither the coefficient comparing A to C nor the coefficient comparing B to C would be significantly different from 0, but that the F-statistic would be significant due to the difference between A and B?