Why is there, seemingly, an interruption in the narrative at Gen 3:20 to say that Adam named Eve?

One clue can be found in the explanation of the name: she is “the mother of all living,” i.e., חַוָּה or Chavvah, which is very close to the verb חֲיָא or chaya, “to live.” Now, apart from “be fruitful and multiply,” there is no mention of reproduction or motherhood in the story of Eve making or before Gen:15-16. At was at that point that the notion that she would give birth at all—in pain or otherwise—was introduced. Once this is introduced, a name that befits her role is given. As with Adam, the Gen 2-3 narrative serves as an origin story explaining the role of women in the world. But this almost makes it sound as if the punishment is having children at all; one need not think so, because the pain of childbirth and the difficulty of raising children that adequate curses for motherhood. Similarly, tilling the earth was something it was implied Adam might have done if he had not fallen, but his particular curse was to do so in sweat and toil.