The Sarum Missal, like our earlier Leofric Missal, labeled this day as In die Circumcisionis Domini, rather than calling it the actual Feast of .... The Roman dialect of the Roman Rite historically referred to it as simply the Octave Day of Christmas.
Sarum/Leofric barely mentioned the Circumcision (only in the Gospel and [Leofric] obiter in a Preface). The theme of the Mass was simply what they called an iterata Solemnitas of Christmas Day. Sarum, despite its enthusiasm for Sequences, fails to mention the Circumcision in its quite lengthy Sequence. The theology that is hinted at is the carnale Commercium whereby His assumption of Flesh and our incorporation into His Divinity are a wondrous exchange (remember the Antiphon in the Divine Office: "O wonderful exhange! The Creator of the human race, taking an ensoulled body, deigned to be born of a Virgin; and, proceeding as a Man without Seed, granted unto us his Godhead." And recall the Christmas Collect which the priest at Mass whispers daily as he mingles wine and water ... almost certainly a felicitous composition by S Leo himself.)
The Mass in the Gregorian Sacramentary and the Missal of S Pius V, celebrated by the Pontiff in Sancta Maria ad Martyres (Agrippa's old foundation of the Pantheon) or, after Gregory IV, in her Basilica trans Tiberim, concentrates on Mary as Theotokos. It is a fitting liturgical custom to take a subsidiary element in a Major festival and to give it a fuller expression nearby; I believe that Byzantium concentrates on our Lady on December 26.
To use January 1 to concentrate on the Lord's Circumcision is an example of the 'historicisation' of the Calendar and, I suspect, a medieval instinct. This does not make it reprehensible. But it does make it rather awkward to sweep a lot of much older liturgical material right away from January 1, replacing it with expressions of the fashions of a later age.
Which is exactly what Cranmer did. As Dom Gregory Dix loved to explain, the Reformers, in their endearing desire to be 'biblical', often revealed themselves as Very Medieval Men! So, in 1549, into the Anglican Divine Office came Genesis 17 and Deuteronomy 10; Romans 2 and Colossians 2. The Epistle became Romans 4. And this was to be the Collect:
Almyghtie God, whiche madeste thy blessed sonne to be circumcised and obedyente to the law for man; graunt us the true circumcision of thy spirite, that our hertes, and al our membres, being mortifyed from al worldy and carnal lustes, may in al thinges obey thy blessed wil; through the same ...
What is the advantage of celebrating the Circumcision? It reminds us that God became a male Man, fully endowed even with those bodily members which Victorian males did not always discuss in front of the Ladies, nor the Ladies in front of the Gentlemen. It reminds us that He was and is, like His Mother and His Apostles, a Jew; it brings before us great swathes of Biblical teaching about Circumcision and Covenant. It commemorates the Lord's first blood-shedding for our Salvation.
Could it go into the Missal as a Votive? I think not, because the convention is that we do not have votives of events in the lives of our Lord and His Mother.
I suggest that it could enter the Liturgy, possibly in a later edition of the Ordinariate Rite, on January 2 as an optional alternative to the celebration that day of our Redeemer's most precious Name.