On the matter of the relationship with Jews, the USCCB commentary says:
14. Does the wider use of the extraordinary form of the rites of Holy Week reflect a change in the Churchs eaching on anti-Semitism?
No. The 1962 Missale Romanum already reflected Blessed John XXIIIs revision of liturgical language often construed as anti-Semitic. In 1965, the watershed statement Nostra Aetate, of the Second Vatican Council then repudiated all forms of anti-Semitism as having no place within Christian life. When Pope Paul VI issued the Missale Romanum of 1969, the only prayer for the Jewish people in the Roman liturgy was completely revised for Good Friday to reflect a renewed understanding of the Jews as Gods chosen people, first to hear the word of God.
Throughout his papacy, John Paul II worked effectively to reconcile the Church with the Jewish people and to strengthen new bonds of friendship. In 1988, Pope John Paul II gave permission for the Mass to be celebrated according the Missale Romanum of 1962 only as a pastoral provision to assist Catholics who remained attached to the previous rites, thereby hoping to develop closer bonds with the family of the Church.
By this new Apostolic Letter, Pope Benedict XVI is merely extending such permission for wider pastoral application, but remains committed to the need to overcome past prejudices, misunderstandings, indifference and the language of contempt and hostility [and to continue] the Jewish-Christian dialogue … to enrich and deepen the bonds of friendship which have developed.
Fr. John Z. (who has some problems with the USCCB’s unofficial translation) adds this:
Third, private Masses/liturgies in the old form can’t be celebrated in private in the Triduum. That is normal and reasonable. that is the way it is in the Novus Ordo. In places where the older form is established in a parish for the older use, the Triduum CAN be celebrated with the older books. However, in parishes where the newer forms are the usual fare, and there is a regularly scheduled Mass with the older form, when the Triduum arrives, the older, extraordinary liturgy must give way to the ordinary. That is logical. In the Novus Ordo, as in the older days, there cannot be two Masses of the Last Supper on Holy Thursday, two Good Friday liturgies, or two Vigils. So, in this case, the ordinary takes precedence.
Fourth, the issue the Triduum and this reasonable restriction has nothing to do with the prayer about the Jews on Good Friday. This is simply a matter of what the Church’s logical practice is based on the sacred nature of those Triduum liturgies. The Jews were not part of the equation. Remember also that the 1962 Missale Romanum is used for the Triduum in those places where it will be permitted to use the older form at that time. The 1962 edition and not some earlier edition before the changes to those Good Friday petitions.
Key points from the Motu Proprio itself:
Art. 2. In Masses celebrated without the people, any priest of Latin rite, whether secular or religious, can use the Roman Missal published by Pope Blessed John XXIII in 1962 or the Roman Missal promulgated by the Supreme Pontiff Paul VI in 1970, on any day except in the Sacred Triduum. For celebration in accordance with one or the other Missal, a priest does not require any permission, neither from the Apostolic See nor his own Ordinary.
Art 10. It is lawful for the local Ordinary, if he judges it opportune, to erect a personal parish according to the norm of canon 518 for celebrations according to the older form of the Roman rite or appoint a rector or chaplain, with due observance of the requirements of law.
From the pope’s letter to the bishops:
There is no contradiction between the two editions of the Roman Missal. In the history of the liturgy there is growth and progress, but no rupture. What earlier generations held as sacred, remains sacred and great for us too, and it cannot be all of a sudden entirely forbidden or even considered harmful. It behooves all of us to preserve the riches which have developed in the Churchs faith and prayer, and to give them their proper place. Needless to say, in order to experience full communion, also the priests of the communities adhering to the former usage cannot, as a matter of principle, exclude celebrating according to the new books. The total exclusion of the new rite would not in fact be consistent with the recognition of its value and holiness.
Note: there is nothing about this being a three year experiment. Rather:
Furthermore, I invite you, dear Brothers, to send to the Holy See an account of your experiences, three years after this Motu Proprio has taken effect. If truly serious difficulties come to light, ways to remedy them can be sought.