" No it wouldn't. "Ο ΩΝ" does not exist in any fashion we comprehend, being the source of existence.
If no one can comprehend God, then they shouldn't be describing Him at all. It appears though, that's not the case. God Himself came to teach who He was. That's where the "Ο ΩΝ"(only begotten) came from. Regardless of the fact, that we exist in His creation, we are in His image, which means we can rationally ponder such things as time, and the consequences of not having time, or a measure of time.
" Indeed, given that "existence" is only something we can comprehend and is a created state, God does not "exist"."
God does exist. He said, "I Am!" I can also read His words, and know exactly where they came from by examination. In order to exist, there must be a timescale to reference such things as eternity, order cause and effect, ponder, decide, act, ect. If there's no time, there is nothing. None of that means that God's clock is the clock of this world, it isn't.
God always had the capacity to extend himself, which is what begotten means here. My point is, that until the Father decided to create and do so, there was no action to beget. John 1:18 No one has ever seen God, but God the One and Only,["Ο ΩΝ", only begotten] who is at the Father's side, has made him known. Only God Himself could be His right hand man. He was made flesh: John 1:14 The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the One and Only,["Ο ΩΝ"] who came from the Father, full of grace and truth.
All these passages identify Jesus as God, and contain action words. The action was the point in time, when God pondered the decision to create. That is where His decision to beget arose in time. His begetting the 2nd person of the Trinity infinitely before His contemplation of the action, makes no sense. The capacity was, but the 2nd person of the Trinity is not capacity, He is a realization of that capacity.
The Nicene Creed contains the words, "before all worlds" which is correct, and it doesn't matter at what point in time that was, as long as it wans't before a decision was made to create. It is not correct to use the word "eternity". Both Gen 1 and John 9 indentify a particular point in time; it is "in the beginning". John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
You are mistranslating "Ο ΩΝ". The phrase "only begotten" is "μονογενις".