Pardon the length, but I don't fit into boxes very easily.
Some thoughts on the "Catholic" side:
I view the apostolic succession in the Anglican churches as every bit as legitimate as that in Catholic and Orthodox churches -- though I think that's uncontroversial among Anglicans.
The Incarnation is central to my spirituality. The sacraments are similarly important to me in worship. God often touches us through the physical world. This includes but is not limited to the seven official sacraments.
The Eucharist is one of the most important ways that God becomes present to us. It should be a part of most of our gatherings for worship as Christians -- certainly every Sunday, barring unusual circumstances, but in many of our other gatherings as well.
I'm uncomfortable with some of the more "Protestant" sections of the 39 Articles, and I don't think I could affirm them as written. I ask the saints for their prayers on occasion, I pray for my loved ones who have died, and I think there may well be something like Purgatory in the afterlife.
I like a lot of the high-church elements of worship: Bowing, kneeling, vestments, incense, chanting, and so on. These actions of my body and senses offer worship in ways that mere words cannot.
Some thoughts on the "Evangelical" side:
Like the Evangelicals, I think it's important for a person to make their own deliberate commitment to follow Jesus and to accept the forgiveness that comes through the work of Christ. I don't want to limit God here -- I think that God's grace can come to all people, even those who don't believe or understand -- but for Christians, following the normal path of the Christian life, it's important to choose for one's self to be Christian, and not just follow along automatically with what the family and community are doing.
Like the Evangelicals, the Bible is very important to me. I don't see the Bible as inerrant (here, I part ways with the Evangelicals), but it has to be studied and wrestled with. Christians should read, study, and know the Bible thoroughly.
An additional personal quirk:
I have a preference for believer's baptism, a holdover from my Baptist days. This puts me at odds with common Anglican practice, though it puts me in agreement with baptism as the rite of full membership in the church. (I understand the church's reasons for infant baptism, and I do welcome baptized infants as full members of the church.)
How does this relate to the 1979 BCP?
Not everything here relates to the 1979 revision. The Episcopal Church has always maintained apostolic succession; that's not new. The Episcopal Church has always read large passages of Scripture in each Sunday service, whether Eucharist or Morning Prayer; that's not new either.
I do appreciate that elements of Catholic and pre-Schism liturgy were incorporated into the new Prayer Book. One of the reasons I came to the Episcopal Church was to be a part of a church that was deeply rooted in history, and our history goes back much further than the English Reformation.
I appreciate that the 39 Articles were placed in an appendix of Historical Documents in the back of the book. They are important as history, but I would not want to see them as binding on us, because of the issues noted above.
Some additional points about the 1979 BCP:
Liturgy in the vernacular was an important insight of the Reformers. Revision of the liturgy into 20th-century English was good and important. The Oxford Guide article mentioned that this was considered especially important in the rites that involve vows (baptism, marriage, confirmation, and ordination). "This is my solemn vow" sends shivers down my spine in a way that "Thereto I plight thee my troth" does not. The 1979 BCP marriage vow has the immense weight of a sacred, sacramental undertaking, exactly because it is written in well-chosen powerful contemporary wording.
Inclusive language means a great deal to me as a woman in the church. I could wish the 1979 authors had gone even a little farther here, but they did well for 1979.
There's a little more grace in the 1979 book. Some places in the 1928 liturgy feel to me like we're grovelling in a self-conscious and artificial way; I noticed it when our parish used Rite I during Lent last year. In the 1979 liturgy, we confess, are absolved, and then having been forgiven we move forward to commune with God. That feels more appropriate to me, and more genuine.
Again, apologies for the length, but these are some of my thoughts about the "new" BCP and how it fits with my own spirituality.