The Preface Dialogue and Preface chanted in Latin in the EF Mass. However, this can and should be chanted in the vernacular in the OF Mass, but many priests simply don't for personal reasons:
Now this priest has no inhibitions, although he once did, about chanting the Preface in the OF Mass, just as he has no inhibitions about looking at the Mass through rose colored vestments and riding in a parade wearing the same color!
Just a bit of a background, the post on which this comment was made was about singing hymns at Mass and that since Vatican II we have borrowed many fine hymns from the Protestant denominations and although these are Protestant, their theology, style and sentiments are "catholic" with a little "c." For example I doubt many Catholic know that "Hark the Herald Angels Sing" is a Wesley hymn.
Nonetheless, hymns technically are not the music of the Latin Rite Mass. Rather, the parts of the Mass are they hymns of our Mass and the Mass as a whole is the hymn of the Church.
For example, our official hymns of the Latin Rite Mass are all prescribed including:
1. The Entrance Chant
2. The Kyrie
3. The Gloria
4. Offertory Antiphon
6. Pater Noster and Doxology
7. Agnus Dei
8. Communion Antiphon
Of the above, the Entrance Chant, Offertory and Communion Antiphons are not fixed but change day to day and truly are variable hymns, but prescribed for each Mass. Also as options for the Rite of Sprinkling Holy Water, the water's blessing may be sung with its corresponding Asperges or Vidi Aquam during Eastertide.
But in addition to that their are "priestly" parts that may be chanted:
1. The Sign of the Cross
2. All greetings including the Preface dialogue
3. The Introduction to the Penitential Act
4. The Absolution to the Penitential Act
5. The Collect, Prayer over the Offerings and Post Communion Prayer
6. The Orate Fratres with the congregation's response
7. The Preface
8. The entire Eucharistic Prayer, any choice
9. The embolism after the Lord's Prayer
10. The priest's prayer, "Lord Jesus Christ..."
11. The Ecce Agnus Dei with the Congregation's response
12. The Solemn Blessing
13. The Blessing and Dismissal
And if that isn't enough singing, The entire Liturgy of the Word may be sung, including the lessons, the Responsorial Psalm, Gospel Acclamation and the Gospel itself!
So, do we really need added on metrical hymns and anthems, either Catholic or Protestant to add to this glorious mix? Not really, but even in Pre-Vatican II times, anthems were sung in addition to the Offertory Antiphon and Communion Antiphons and I believe hymns could be sung at the Procession and recessional even at a High Mass without deacon or sub-deacon, but not perhaps at a Solemn Sung Mass (with deacon and sub-deacon) but I'm not sure to be honest.
But back to the original question as to why priests don't chant all the parts that are allowed to be chanted by the priest in the Ordinary Form of the Mass?
My humble opinion:
I don’t know if the psychology of the priest facing the congregation during Mass impacts this, but since I’ve been celebrating the EF High Mass and just this past Monday night an OF Mass ad orientem where I chanted the entire preface and Eucharistic Prayer II, I find that when I don’t see the congregation in front of me as though I’m performing before them, but rather have them behind me as though I’m a part of them, I lose a great deal of my inhibitions especially as it concerns my ability to chant well.
In other words, I feel that I have to make it a good performance for my "audience" when I’m chanting toward them--that's the psychology of facing people and singing to them. That’s not the case,though, when I’m chanting “with them” facing the same direction as they are and leading them in Liturgical chanted prayer.
But in terms of the High Mass of the Extraordinary Form, the rubric states what you are to sing and what you aren’t. In the Low Mass nothing of the Mass is sung, but four hymns are allowed. Thus, the Ordinary Form Mass borrowed heavily from the EF's Low Mass allowance for hymns in the vernacular although the Mass itself was not sung one bit. The so-called renewal of singing the Mass in the Ordinary Form relied entirely upon a "corrupt" pre-Vatican II allowance for hymns to be sung at a Low Mass (usually Marian and other saints devotional hymns and Benediction hymns. Whereas the authentic post-Vatican II renewal of the Mass should have focused on the laity singing the actual parts of the Mass including the official Entrance Chants and Offertory and Communion chants. That did not take place and still hasn't for the most part some 50 years after Vatican II began! That's a shame!
In the High Mass, the priest is mandated to chant the greetings, collect, Gospel, Preface dialogue and preface (not the Roman Canon as this is in low voice) and the Pater Noster, the Dismissal and Blessing. It’s not the priest's choice in other words. Whereas in the OF sung Mass, the priest can pick and choose as there is no distinction between low and high Mass, just Mass with music or without music or with some music depending on how the priest feels that day. That's a very odd criteria and so "chanting priests" are becoming as rare as Pink (I mean rose) Flamingos!