Actually this is exactly what makes it not true according to the majority of text. You accept it only to support the theology you believe. Typical Calvinist do this all the time. Making scripture support doctrine.
1Pe 1:3 Blessed be the God and Father of our Lord Jesus Christ, which according to his abundant mercy hath begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead,
MB
This argument is also made toward those who are non-calvinists.
So, it really is not supportable.
the principle of both the words remains that it took a direct act of God to make it happen. He caused the lively hope, is as valid as He has begotten us. For both focus upon the very direct act of God.
Straining over the use of one word over the other is of little value.
The verse could also use: constructed, created, or any word other than one suggesting remodeling or remaking of the old.
I like the words "born again into a lively hope" because it is so very very easy on the tongue.