I don’t see the thinking behind the assertion. Immortal in essence means that they could not die. As mentioned above they A+E were not initially given immortality. God created them to be eternal beings, not immortals. They had the opportunity to cooperate in free will with grace =Tree of Life and would have reached immortality. Had they been immortal then of what difference would they have been than the angels-demons condemned. They would have been condemned to a state they never existed in as mortal?
The Catechism give the clear teaching of the Church on the immortal soul, and
bodily death for which man was not destined, but that entered the world through sin.
Catechism of the Catholic Church
365 The unity of soul and body is so profound that one has to consider the soul to be the “form” of the body:234 i.e., it is because of its spiritual soul that the body made of matter becomes a living, human body; spirit and matter, in man, are not two natures united, but rather their union forms a single nature.
234 Cf. Council of Vienne (1312): DS 902.
366 The Church teaches that every spiritual soul is created immediately by God - it is not “produced” by the parents - and also that it is immortal: it does not perish when it separates from the body at death, and it will be reunited with the body at the final Resurrection.235
235 Cf. Pius XII,
Humani Generis: DS 3896; Paul VI,
CPG § 8; Lateran Council V (1513): DS 1440.
1008 Death is a consequence of sin. The Church’s Magisterium, as authentic interpreter of the affirmations of Scripture and Tradition, teaches that death entered the world on account of man’s sin. 569 Even though man’s nature is mortal God had destined him not to die. Death was therefore contrary to the plans of God the Creator and entered the world as a consequence of sin. 570 “Bodily death, from which man would have been immune had he not sinned” is thus “the last enemy” of man left to be conquered. 571
569 0Cf. Gen 2:17; 3:3; 3:19; Wis 1:13; Rom 5:12; 6:23; DS 1511.
570 Cf. Wis 2:23-24.
571 GS 18 # 2; cf. 1 Cor 15:26.