You are getting close on the begotten thing but no cigar just yet.
There are a couple of different words translated as begotten in the KJV. The most common is monogenes, and so we see it in such verses as John 3:16 in reference to Jesus and so we memorized it as his only begotten son.
Thing is, there's actually a problem with translating it that way, the primary one being it's not quite right. It adds something. For instance those who say first there was the Father, and then he begot Jesus, such as the Jehovah's Witnesses, hang a lot onto that translation.
Here's the example in the KJV that shows it to be wrong.
Heb 11:17 KJV
(17) By faith Abraham, when he was tried, offered up Isaac: and he that had received the promises offered up his only begotten son,
See the problem? Isaac was not Abraham's only begotten son, never was, Ishmael was begotten by Abraham and would, for a time, have been his only begotten.
This verse is actually one of the ones Muslims look at and conclude the Jews changed the scriptures, and why they say it was Ishmael that was going to be sacrificed. (Just an aside)
Now if we are going to do it rightly, there is indeed one word that translates monogenes very accurately, it would be "unique" or "one of a kind".
The common use of "only" today doesn't really go far enough and leaves itself open to misinterpretation again, as you point out. I think it must be the situations that people speak of an only child as the foundation of that translation.
Now there is one translation I know of that actually has had the guts to go all the way to a correct rendering of monogenes as "unique" and they've been soundly criticised for it. That would be the International Standard Version, ISV.
Heb 11:17 ISV
(17) By faith Abraham, when he was tested, offered Isaac. The man who had received the promises was about to offer his unique son,
See how nicely it works. Isaac was not the only begotten; however, he indeed was unique. He was the one promised son.
And so too Jesus is unique, one of a kind, the promised son.
As far as I can tell the translational problem goes back to about the year 400 when the Latin form of the Nicene used "unigenitum" for monogenes, which if you translated unigenitum it indeed would be only begotten. And Jerome used unigenitum for monogenes in the Latin Vulgate translation and that seems to be the reason "only begotten" has had so much momentum for all these centuries.
It goes back to a couple related words and we can even see the different roots in English.
For instance we see genus, which is refering to kind. Monogenes, really is simply mono meaning one, and a root meaning kind, one of a kind.
The mistaken root is seen in generation. So when you generate something you produce it and that's what only begotten would mean, one generated.
The Nicene creed we see use the monogenes and then try to instantly correct the wrong interpretation with the before all worlds or before time phrase. Jesus was not begotten as you or I were begotten.
Anyway translate monogenes as unique and a very interesting thing happens. The references to Jesus being begotten come down to a couple, like when he was raised from the dead, but there really isn't any referring to before the incarnation.
So anyway, while you have some pretty good reasoning going your way, I think you just haven't become aware of the whole issue. It's really easy when we see the liberals and their desire to throw out scripture and substitute themselves as the authorities to go too far in what we hold onto. I would hold onto scripture but I wouldn't specifically hold onto the KJV it's an imperfect translation too as all are.
