It could not be the Holy Spirit, as the Holy Spirit doesn't have a body (beyond the Son's, more on that later).
It could not be the Father, since God is not bound by death. (God is eternal: always is, was and shall be, as taught by the Catholic Church).
Who does that leave? The Son.
The Doctrine of the Trinity holds that the Triune God is three persons sharing one essence/one substance (the Son being "consubstantial" with the Father is in the Nicene Creed) -- only one of whom had a mortal/fleshy nature to give up in sacrifice.
(CCC = Catechism of the Catholic Church)
198 Father is the first divine person of the Most Holy Trinity
430 Jesus means in Hebrew: "God saves." At the annunciation, the angel
Gabriel gave him the name Jesus as his proper name, which expresses
both his identity and his mission. Since God alone can forgive sins,
it is God who, in Jesus his eternal Son made man, "will save his
people from their sins". In Jesus, God recapitulates all of his
history of salvation on behalf of men.
The Second Person is revealed in the flesh: why? (CCC 458-459)
458 The Word became flesh so that thus we might know God's love: "In
this the love of God was made manifest among us, that God sent his
only Son into the world, so that we might live through him."(1 John 4:9) "For
God so loved the world that he gave his only Son, that whoever
believes in him should not perish but have eternal life." (John 3:16)
459 The Word became flesh to be our model of holiness: "Take my yoke
upon you, and learn from me." Mt 11:29; ⇒ "I am the way, and the truth, and the life; no one comes to the Father, but by me." Jn 14:6. "On the mountain of the Transfiguration, the Father commands: "Listen to him!" (Mark 9:7) Jesus is the model for the Beatitudes and the norm of the new law: "Love one
another as I have loved you." (John 15:12) This love implies an effective
offering of oneself, after his example.
684 Through his grace, the Holy Spirit is the first to awaken faith in
us and to communicate to us the new life, which is to "know the Father
and the one whom he has sent, Jesus Christ." But the Spirit is the
last of the persons of the Holy Trinity to be revealed. St. Gregory of
Nazianzus, the Theologian, explains this progression in terms of the
pedagogy of divine "condescension":
The Old Testament proclaimed the Father clearly, but the Son more
obscurely. The New Testament revealed the Son and gave us a glimpse of
the divinity of the Spirit. Now the Spirit dwells among us and grants
us a clearer vision of himself. It was not prudent, when the divinity
of the Father had not yet been confessed, to proclaim the Son openly
and, when the divinity of the Son was not yet admitted, to add the
Holy Spirit as an extra burden, to speak somewhat daringly.... By
advancing and progressing "from glory to glory," the light of the
Trinity will shine in ever more brilliant rays.
685 To believe in the Holy Spirit is to profess that the Holy Spirit
is one of the persons of the Holy Trinity, consubstantial with the
Father and the Son: "with the Father and the Son he is worshipped and
glorified." For this reason, the divine mystery of the Holy Spirit
was already treated in the context of Trinitarian "theology."
Put another way: of the three persons in the Trinity, only one had the characteristic of being mortal/human that could giving up a life for any reason. (The Salvation of Mankind).
The other point is less canonical and more practical, and involves setting an example as a form of sending a message.
From First to Last ...
Beginning/Baptism: Jesus was already divine (being of both divine and human nature, per Catholic Dogma and most Christian belief) so why did he go into the River Jordan to let John the Baptist baptize him? Jesus, didn't need to repent from sin as those who were coming to John's baptisms need to. Jesus set by his actions the perfect example for how to turn away from sin and back towards God. (No further digression into Baptism, as that isn't the question).
Middle/Ministry: Jesus' ministry was about teaching by word and deed how to turn back towards God, and how to better fulfill the role of being God's children. About a thousand years later, St. Francis provided the following advice to all Christians in terms of how to be a better disciple: "Preach the Gospel every day. Use words when you have to."1
Jesus didn't just talk the talk, he walked the walk. His ministry as recorded in the Gospels demonstrated (among other things) how to live correctly (as God had intended for us to).
The crucifixion can be seen through the lens of Jesus' command to his disciples: "love one another as I have loved you." Love in this sense is best understood as agape -- which is a self-sacrificial love, a divine love, and a love that can hurt. Jesus set the extreme example of how much of a sacrifice one man can make for his fellow man by giving it all up, to include His life.
That example only works as a message (actions speaking louder than words) if you are made of flesh in the first place. If you are a being made wholly of spirit, you can't sacrifice a mortal life, and thus can't set this example.
The message obviously got across. (Pun intended).
1 The provenance of that aphorism has been challenged, but it is consistent with St. Francis' message.