Did Jerome Believe in the Eucharist?

What you cite is not actually from St. Jerome but a writing of Church Father Origen that St. Jerome translated. Origen undoubtedly had his doctrinal problems, and yet he also wrote very well on other subjects, including the Eucharist.

In the text you cite, Origen actually affirms the reality of the Eucharist. He does so in comparing how the faithful rightfully respond when a eucharistic crumb falls to the floor by pointing out that they need to have similar reverence when God’s living word is preached to them.

The following excerpt is taken from Pope John Paul II’s weekly General Audience on June 5, 2002:

Origen, who comments on our Psalm in one of his homilies, translated and disseminated by St. Jerome in the West, actually interweaves the word of God with the Eucharist: “We read the Holy Scriptures. I believe that the Gospel is the Body of Christ. I believe that the holy Scriptures are his teaching. And when he says: he who eats my flesh and drinks my blood (John 6:53), although these words can also refer to the [eucharistic] mystery, yet the Body and Blood of Christ is truly a word of Scripture, the teaching of God. When we are about to receive the [eucharistic] mystery, if even a tiny crumb falls, we feel lost. When we are listening to God’s word, when our ears perceive the word of God and the body and blood of Christ, what great danger would we not fall into were we to think about something else?” (74 Omelie sul Libro dei Salmi [74 Homilies on the Book of Psalms], Milano 1993, pp. 543-544).