As I prefer the “extraordinary” Use to the “ordinary” Use for my own spiritual sustinance, I am happy to read that the Holy Father uses it in his private chapel. As Patriarch of the West, he has every right to use any western Rite. As Supreme Pontiff, he has every right to use any approved Rite, Eastern or Western. One of the traditional practices of a Solemn Papal Mass in the 1962 and prior rite was having the Pope attended by not the usual one deacon and one sub-deacon, but two, one Latin and the other Greek. Both would chant the Epistle and the Gospel, one of them in Greek and the other in Latin, in order to show both the equality of East and West as valid expressions of the Catholic Faith, and as an expression of the Popes authority over the entire Body of Christ, as the Vicar of Christ.
There are many photos of Eastern Rite Bishops offering the Eastern Rite Masses (or Divine Liturgies) both at the Second Vatican Council and prior to the Council. There have even been Greek Popes who offered the Eastern Liturgy at St. Peters, (both the present building and the two that existed prior to the present one) These Popes also learned how to offer the Western Rites.
Being Bi-Ritual is not anything new. One of my favorite photos of the late Archbishop Sheen shows him vested in Eastern Rite vestments in order to celebrate the Eastern Divine Liturgy. I know a few Bi-Ritual priests and they benefit from celebrating in both the Eastern and Western Rites. As Pope John Paul II, and many others over time have commented, the Western and Eastern Church are like breathing through both lungs.