Yet the term “begotten” is not used in those cases. God did not father those children, men did.
That presents a bunch of different problems then. Here is the translation you used
"I will tell of the decree; The Lord said to me, "You are My son; this day have I begotten you."
First, Psalms 2:7 has a first and second person:
I (the speaker, David) will tell of the decree [by God]
The Lord said to me (the speaker, David)
"You are my son; this day I have begotten you"
If you want to see a biological and literal notion of "begotten" then you should also consider that the birthing happened ON THAT DAY so it cannot refer to anyone not born on that day. And, if you want to read it literally, it seems to mean that God birthed THE SPEAKER. Is that your understanding?
You are now focused on the verb "y'lee'd'teecha" from the verb y-l-d (to give birth).
Men can't give birth, so God would be saying that he is the MOTHER, not father. In biblical text, though, the word is used sometimes to indicate the father's involvement, but since it is used for both mother and father (it can't be literal birthing and) it doesn't make clear if God is the father or mother. Since it says elsewhere that God is father to a variety of people and things, that doesn't clarify the use in this case. So that's another problem.
Next, that verb doesn't just mean literally birthing (which is why it can refer to fathers). In Zephania 2:2, the same verb is used to indicate that laws are "born". A day "births" things in Proverbs 27:1. In Ecclesiastes 4:14, a man becomes poor (using the same verb for "becomes").
Take a look at Deut 2:18 -- in that verse, either a rock gave birth to the nation (literally?) or God, symbolized by the rock "begot" the nation -- same verb.
Isaiah 33:11, using that verb, someone literally birthed straw?
Isaiah 66:8 -- a nation is born using that word.
So the word can mean "biologically gave birth to" and also symbolically, "brought about" and it is used about a variety of "birthers" and "things being born."
In 2:7, you are trying to draw out a biological relationship because of the use of the word, but then are missing what the word is used for in a variety of biblical contexts so your interpretation is limited.
Remember when I gave you this translation?
"I will tell of the covenant of the Lord. He said: “You are as dear to Me as a son to a father, pure as if this day I had created you.” "
Now you can see why it might be more complete and useful for understanding than just using the Judaica Press one.