Scripture-first response
“God sending His Son proves the Son existed as a separate person.”
Scripture also speaks of prophets and John as sent in relation to their mission. Galatians immediately says the sent Son was made of a woman.
What the objection is claiming
Sending language is assumed to describe spatial travel from one divine person to another location.
Galatians 4:4
But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law,Open full referenceGod sent forth His Son as one made of a woman and born under the law.
How the passages answer it
Sending often names authorization, commissioning, and arrival in mission. The Son is sent as one made of a woman. Eternal preexistence belongs to God and His Word; Sonship is the incarnate form that enters history.