As you know, “soul of the Church” terminology has been in vogue for some time. Msgr. Fenton demonstrated well how that concept is erroneous.
Certainly the hierarchy was already in place after Out Lord appointed St. Peter the Pope and consecrated the Apostles as bishops. Also, He issued the mandate for the Sacrament of Baptism after His Resurrection. I think the notion that the Church was born on Pentecost is a bit imprecise. Perhaps the analogy might be between conception and birth, where the baby, already alive, becomes manifest. Babies don’t come into existence at birth but merely come out into the world for all to see.