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I think that the argument would go essentially like this. "Person" is a "who"; "nature" is a "what". Christ (as you correctly note) is not a divine person "stuck to" a human person. There is only one "who," one person, in Christ.

Having said that, the question then becomes "who is that 'who' in Christ". "And you ... who do you say that I am?"

To that question, the primary answer we must give is "You are the Christ, the Son of the Living God, the incarnate Second Person of the Blessed Trinity, God the Son". That identifies a divine person, not a human person.

Christ was the same person before the Incarnation as after. Before the Incarnation, there is no question that he did not have a human nature. After the Incarnation, there is no question that he remains Divine. Since he was a Divine person before the Incarnation, he remains a Divine person today.

The correct formulation, in the opinion of most theologians AFAIK, that Christ is a Divine person possessed of both a divine nature (obviously) and a human nature.