Why did the Messiah have to be both fully God and fully man?

The short answer

Because the new covenant assigns its mediator a work only God can do and a death only a human can die. One person, both roles, or it cannot be cut.

The full answer

Most of us were handed "fully God and fully man" as a formula to recite at Christmas, never a thing to understand, so asking why is not doubt. It is the beginning of actually believing it. And the instinct underneath the question is right: if "God" is just a label we paste onto a good rabbi, the claim is hollow and you are correct to press it. You may also feel the weight of Israel's confession that God is one and not a man, and you should. Holding that is faithfulness.

So set the Greek vocabulary aside for a moment, the substance and the two natures and the councils, because reciting it has never once satisfied a person who genuinely asked why. The answer was never in Athens. It is in the covenant. Jeremiah 31 promises two things only God can do, write Torah on a human heart and put His own breath inside a person, and it is sealed in blood, which requires a faithful human party, the very thing Adam was not and Israel never managed to be. The work belongs to God. The blood belongs to a man. One covenant, cut in one act, needs one person who is both. Not a riddle to swallow. A requirement to see.

Do not take it from me. Read Jeremiah 31:31-34 and ask who could possibly do what it promises, then read Luke 22:20, where Yeshua (Jesus) lifts the cup and calls it that covenant in His own blood. Hold the two side by side and watch what kind of person the text demands.