"Suppose that N and K are minimal normal subgroups of the primitive group G. Then each of N and K is transitive; but they centralise one other, and so each is semiregular, and so regular. But then it is not possible for there to be a third minimal normal subgroup."
Is the last statement true because there are only two regular representations of a group - the right and left? If so, what is the reason there can only be two regular representations?
There is a result that the centralizer of the image of the left regular permutation representation of a group $G$ is equal to the image of the right regular representation, and the claim follows from this, which is related to the fact that there are only two regular representations.
So, if there are two minimal normal subgroups then they must be the images of the left and right regular representations of some group, which must be the direct product of copies of a nonabelian simple groups.
The smallest such example has degree 60 with $N \cong K \cong A_5$.