The account of John 3 is so well known that, by the time any student is in seminary or Bible college learning Greek, they are almost reciting the passage by memory as they recognize certain words in their translation. It is difficult to think beyond the boundaries of such a well known translation of such a well known passage.
Even if we concede that the translation of modern scholars is strongly influenced by how they memorized a passage in their youth, the question simply moves backward a few centuries until we find the very first translator within some particular language who translated "genneithei anothen" as "born again" rather than "begotten again." What was the reason for that translational decision?
The origin of this error can be appreciated by looking at Nicodemus question to Jesus. In verse 4, Nicodemus asks, "How can a be born (or 'begotten') when he is old? he cannot enter a second time into his mother's womb and be born, can he?" On an intuitive level, to "enter in" (eiselthein, from the Greek root eiserchomai), is the mirrored reflection of "exiting" or being born. This intuitive corollary serves as a powerful first impression, and could easily color the way any of us might understand this passage on first impression.
But this need not be the actual meaning of Nicodemus' question. Although one is certainly born from the womb of their mother, they are also begotten in the womb of their mother. When considered in these terms, there is no fundamental reason that Nicodemus' question favors one translation over the other. Within this limited context, both "born" and "begotten" are equally reasonable. All we know for certain is that Nicodemus used the term "gennao" in his responsive question to Jesus, because this is the word that Jesus used. Ultimately, we do not know for certain which concept was in the mind of Nicodemus when he asked this question.
Moreover, even if we assume that Nicodemus' question was was framed with respect to the process of birth rather than the miracle of new life implanted at the time of begetting, this would prove nothing. The only thing we know with certainty about Nicodemus' frame of mind is that Nicodemus was deeply confused by Jesus teaching. In view of this confusion, it would seem illogical to determine the meaning of Jesus' words by analyzing a sentence of a man who is plainly confused! This is hardly solid exegetical technique, and demonstrates a lack of understanding between narrative literature and epistolary literature.
Apart from Nicodemus' question, there is nothing in the context that would favor the translation "born again" over the translation "begotten again." Moreover, as noted in the previous chapter, the discussion of regeneration in 1st John 3:9 uses the word "sperma." In 1st John 3:9, gennao plainly refers to the male act of begetting. In this case, to translate gennao as "born" would be transparently wrong. We are not "born" of sperma. A new life is "begotten" of sperma. We are born nine months after the implantation of that new life.