Some translations do translate it similar to what you have asked, such as:
American Standard Version
No man hath seen God at any time; the only begotten Son, who is in the
bosom of the Father, he hath declared him .
English Revised Version
No man hath seen God at any time; the only begotten Son, which is in
the bosom of the Father, he hath declared him.
Young's Literal Translation
God no one hath ever seen; the only begotten Son, who is on the bosom
of the Father -- he did declare.
These are the words of interest that are in the verse:
[but the] one and only Son,
μονογενὴς (monogenēs)
Adjective - Nominative Masculine Singular
Strong's Greek 3439: Only, only-begotten; unique. From monos and ginomai; only-born, i.e. Sole.
[ who is Himself ] God
Θεὸς (Theos)
Noun - Nominative Masculine Singular
Strong's Greek 2316: A deity, especially the supreme Divinity; figuratively, a magistrate; by Hebraism, very.
has made [Him] known.
ἐξηγήσατο (exēgēsato)
Verb - Aorist Indicative Middle - 3rd Person Singular
Strong's Greek 1834: To lead, show the way; met: I unfold, narrate, declare. From ek and hegeomai; to consider out, i.e. Rehearse, unfold.
From this, I think that "only begotten God" is viable. "Revealed" conveys a similar meaning, however, the NASB Translation uses the word in this way: explained (2), relate (2), related (1), relating (1).
The billmounce greek-dictionary defines exegeomai:
Definition:
to be a leader; to detail, to set forth in language; to tell, narrate, recount, Lk. 24:35; Acts 10:8; to make known, reveal, Jn. 1:18; Acts 15:12, 14; 21:19*
A different word for revealed may have been:
(See Matthew 10:26)
will not be uncovered,
ἀποκαλυφθήσεται (apokalyphthēsetai)
Verb - Future Indicative Passive - 3rd Person Singular
Strong's Greek 601: To uncover, bring to light, reveal. From apo and kalupto; to take off the cover, i.e. Disclose.
601 apokalýptō (from 575 /apó, "away from" and 2572 /kalýptō, "to cover") – properly, uncover, revealing what is hidden (veiled, obstructed), especially its inner make-up; (figuratively) to make plain (manifest), particularly what is immaterial (invisible).
(See also John 17:6)
I have revealed
Ἐφανέρωσά (Ephanerōsa)
Verb - Aorist Indicative Active - 1st Person Singular
Strong's Greek 5319: To make clear (visible, manifest), make known. From phaneros; to render apparent.
5319 phaneróō (from 5457 /phṓs, "light") – properly, illumine, make manifest (visible); (figuratively) make plain, in open view; to become apparent ("graspable").