Why did the phrase "He begets not, nor is He begotten" appear in Surat Al-Ikhlas in this order, and not "He is not begotten, nor does He beget," considering that the latter order might be grammatically better?

The reason for prioritizing (He begets not) over (nor was He begotten) is that the former is more significant. This is because the misguided among the Arab polytheists, Jews, and Christians attributed a son to Allah, while no one attributed a parent to Allah. So, He began with what is more important, which is negating the begetting of offspring, then alluded to the proof by saying: (nor was He begotten). This is because the evidence for the impossibility of "sonship" is our agreement that He was not begotten by another.