Yes, μονογενὴς whether it's with θεὸς, υἱός, or by itself (see text commentary at end) refers to the Son, not the Father. Where that is obvious is among the textual critics. The textual critics on the committee for the NA28 decides based on objective textual evidence, as seen in the commentary at the end. However, some textual critics who claim to see theological development in the New Testament claim that μονογενὴς θεὸς is impossible because it is too advanced a theology for John to see Jesus as God. So, yes this textual variation does see Jesus as divine at the very least, and even without the article, μονογενὴς θεὸς sure looks like a claim for Jesus to be God, the cause of that particular textual critic's objection.
While the etymology (separating the compound word) of μονογενής is only begotten, the word actually means unique, one of a kind. See the following verse in which Isaac was not the only begotten son. He was one of a kind.
By faith Abraham, when he was tested, offered up Isaac, and he who had received the promises was in the act of offering up his only son [τὸν μονογενῆ], 18 of whom it was said, “Through Isaac shall your offspring be named.” (Heb. 11:17–18, ESV)
Lexicons
μονογενής (monogenēs), ές (es): adj.; ≡ Str 3439; TDNT 4.737—LN 58.52 unique, only, one and only, i.e., one of a kind: (many versions) only begotten (Lk 7:12; 8:42; 9:38; Jn 1:14, 18; 3:16, 18; Heb 11:17; 1Jn 4:9+; Jn 1:34 v.r.) -- Dictionary of Biblical Languages with Semantic Domains: Greek (New Testament)
58.52 μονογενής, ές: pertaining to what is unique in the sense of being the only one of the same kind or class—‘unique, only.’ τὸν υἱὸν τὸν μονογενῆ ἔδωκεν ‘he gave his only Son’ Jn 3:16; τὸν υἱὸν αὐτοῦ τὸν μονογενῆ ἀπέσταλκεν ὁ θεός ‘God sent his only Son’ 1 Jn 4:9; τὸν μονογενῆ προσέφερεν ὁ τὰς ἐπαγγελίας ἀναδεξάμενος ‘he who had received the promises presented his only son’ or ‘… was ready to offer his only son’ He 11:17. Abraham, of course, did have another son, Ishmael, and later sons by Keturah, but Isaac was a unique son in that he was a son born as the result of certain promises made by God. Accordingly, he could be called a μονογενής son, since he was the only one of his kind. -- Louw, J. P., & Nida, E. A. (1996). Greek-English lexicon of the New Testament: based on semantic domains (electronic ed. of the 2nd edition., Vol. 1, p. 590). New York: United Bible Societies.
μονογενής, ές (Hes.+; LXX; Joseph.; loanw. in rabb.) only (so mostly, incl. Judg 11:34; Tob 3:15; 8:17) of children: of Isaac, Abraham’s only son (Jos., Ant. 1, 222)Hb 11:17. Of an only son (Plut., Lycurgus 31, 8; Jos., Ant. 20, 20)Lk 7:12; 9:38. Of the daughter (Diod. S. 4, 73, 2) of Jairus 8:42.—Also unique (in kind) of someth. that is the only example of its category (Cornutus 27 p, 49, 13 εἷς κ. μονογενὴς ὁ κόσμος ἐστί. μονογενῆ κ. μόνα ἐστίν=‘unique and alone’; Pla., Timaeus 92C). Of the mysterious bird, the Phoenix 1 Cl 25:2.—In the Johannine lit. μ. is used only of Jesus. The mngs. only, unique may be quite adequate for all its occurrences here (so M-M., RSV et al.; DMoody, JBL 72, ’53, 213-19; FCGrant, ATR 36, ’54, 284-87). But some (e.g. WBauer, Hdb.) prefer to regard μ. as somewhat heightened in mng. in J and 1 J to only-begotten or begotten of the Only One, in view of the emphasis on γεννᾶσθαι ἐκ θεοῦ (J 1:13 al.); in this case it would be analogous to πρωτότοκος (Ro 8:29; Col 1:15 al.). τὸν υἱὸν τὸν μ. ἔδωκεν J 3:16 (Philo Bybl. [100 AD] in Euseb., Pr. Ev. 1, 10, 33: Cronus offers up his μονογενὴς υἱός). ὁ μ. υἱὸς τοῦ θεοῦ vs. 18; cf. J 1:34 v.l. τὸν υἱὸν τὸν μ. ἀπέσταλκεν ὁ θεός 1J 4:9; cf. Dg 10:2. On the expr. δόξαν ὡς μονογενοῦς παρὰ πατρός J 1:14 s. Hdb. ad loc. and PWinter, Zeitschrift für Rel. u. Geistesgeschichte 5, ’53, 335-65 (Engl.). Cf. also Hdb. on vs. 18 where, beside the rdg. μονογενὴς θεός (considered by many the orig.) an only-begotten one, God (acc. to his real being), or a God begotten of the Only One, another rdg. ὁ μονογενὴς υἱός is found. MPol 20:2 in the doxology διὰ παιδὸς αὐτοῦ τοῦ μονογενοῦς Ἰησοῦ Χριστοῦ.—On the mng. of μονογενής in history of religion cf. the material in Hdb.3 25f on J 1:14 (also Plut., Mor. 423A Πλάτων. . . αὐτῷ δή φησι δοκεῖν ἕνα τοῦτον [sc. τὸν κόσμον] εἶναι μονογενῆ τῷ θεῷ καὶ ἀγαπητόν; Wsd 7:22 of σοφία: ἔστι ἐν αὐτῇ πνεῦμα νοερὸν ἅγιον μονογενές.—Vett. Val. 11, 32) as well as the lit. given there, also HLeisegang, Der Bruder des Erlösers: Αγγελος I ’25, 24-33; RBultmann J, 47, 2; 55f; FBüchsel, TW IV 745-50. -- Arndt, W., Gingrich, F. W., Danker, F. W., & Bauer, W. (1979). A Greek-English lexicon of the New Testament and other early Christian literature : a translation and adaption of the fourth revised and augmented edition of Walter Bauer’s Griechisch-deutsches Worterbuch zu den Schrift en des Neuen Testaments und der ubrigen urchristlichen Literatur (p. 527). Chicago: University of Chicago Press.
Here is Bruce Metzger's comments from A Textual Commentary on the Greek New Testament, Second Edition
1:18 μονογενὴς θεός {B}
With the acquisition of 𝔓66 and 𝔓75, both of which read θεός, the external support of this reading has been notably strengthened. A majority of the Committee regarded the reading μονογενὴς υἱός, which undoubtedly is easier than μονογενὴς θεός, to be the result of scribal assimilation to Jn 3:16, 18; 1 Jn 4:9. The anarthrous use of θεός (cf. 1:1) appears to be more primitive. There is no reason why the article should have been deleted, and when υἱός supplanted θεός it would certainly have been added. The shortest reading, ὁ μονογενής, while attractive because of internal considerations, is too poorly attested for acceptance as the text. Some modern commentators4 take μονογενής as a noun and punctuate so as to have three distinct designations of him who makes God known (μονογενής, θεός, ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς …).
[It is doubtful that the author would have written μονογενὴς θεός, which may be a primitive, transcriptional error in the Alexandrian tradition (Υς/Θς). At least a {D} decision would be preferable. A.W.]