The Greek word monogenes has nothing to do with begetting but means one of a kind unique
The Lexical Meaning of "Monogenes"
(Monogenes" (from, monos=only, single; and, genos=kind. Lit., of a single kind; unique)
"Lit. it means 'of a single kind', and could even be used in this sense of the Phoenix (1 Clem.25.2). It is only distantly related to gennao, beget. The idea of 'only begotten' goes back to Jerome who used unigenitus in the Vulg. to counter the Arian claim that Jesus was not begotten but made" (Colin Brown, Ed., The New International Dictionary of New Testament Theology, Vol.II, p. 725)
"Single of its kind, only; used of only sons or daughters...used of Christ, denotes the only son of God or one who in the sense in which he himself is the son of God has no brethren" (J H Thayer; Greek-English Lexicon, p.417)
"only...Also unique (in kind) of someth. that it the only example of its category...'unique and alone'" (W F Ardnt and F W Gingrich; A Greek-English Lexicon, p.529)
"the only member of am kin or kind: hence, generally, only, single, unique" (H G Liddell and R Scott; A Greek-English Lexicon, p.1144. Revised Edition)
"Monogenes, is literally “one of a kind,” “only,” “unique” (unicus), not “only-begotten,” which would be monogennetos, (unigenitus), and is common in the LXX in this sense" (J H Moulton & G Milligan; Vocabulary of the Greek New Testament. pp. 416-417)
" The Usage outside the NT. In compounds with genēs, adverbs describe the nature rather than the source of derivation. Hence monogenēs is used for the only child. More generally it means “unique” or “incomparable.” The LXX has the first sense in Judg. 11:34 and the second in Ps. 22:20. agapētós occurs in Gen. 22:2, 12 where monogenḗs might have been used (cf. Mk. 1:11), but while the only child may be “beloved,” the terms are not synonymous." (G Kittel and G Friedrich, Theological Dictionary of the New Testament, p. 607. single vol ed)