Priesthood service is a visible position of authority, honor, and respect. I have three examples that doctrinally do not require the priesthood. 1) The bishopric sits on the stand. I understand this is in the handbook. Not doctrinally required but emphasizes the visible position of authority, honor, and respect. Even when an individual outside of the bishopric presides, the bishopric still sit on the stand. 2) The YM pass the sacrament. Again there is not a doctrinal reason for this. Passing the sacrament is not technically a ritual or priesthood function. It is a practical matter of getting the bread and water to everyone in the congregation. Yet we have cordoned off this task as something that only priesthood holders may perform. We also have placed strong guidelines on the "uniform" or color of shirt to be worn when doing this task. 3) Until very recently, only priesthood holders were allowed to act as witnesses. For my temple wedding the fathers on both sides acted as the witnesses and it was a visible position of honor and respect.
So for practical purposes - we could cordon off any important visible task and say that only priesthood bearers may perform that task. Speaking in SM, teaching in SS, offering public prayers, these and many other tasks can be and have historically been (at some time or another by some group or another) reserved for men (and men with certain levels of authority).
Yes, this is strange. I am hopeful that this is a precursor to giving women the priesthood (as in visible roles of authority, honor, and respect) some time in the future. DW told me about this after the women's session. She told me that Pres. RMN had said that women already have access to priesthood power. My immediate thought was - "That's great. What do they get to do with this power?" She responded that women could do things like pray and receive revelations for their family. So essentially, Pres. RMN is saying that women use their access to priesthood power to do things that we never thought of as requiring priesthood power before.
It can be like walking. Almost everyone walks and it is an unremarkable thing. Doctrinally the case can be made that we use the priesthood to walk. The priesthood according to LDS definition is the power of God delegated to man. It is God's power that organizes and maintains matter in a useful form. It is God's power through the "breath of life" and light of Christ that powers all living things. Therefore, when you send messages to your legs to walk and they walk you it could be said technically that you are using priesthood power to do so. Congratulations!
Looking back at the talk of Pres. RMN, he also says that the authority by which women in the church perform their callings is also priesthood authority. "You are given priesthood authority to function in that calling." This again seems to go back to my example about walking, "You didn't know that you were already using priesthood power to do regular ordinary stuff every day!" Yet it completely sidesteps the elephant in the room - that we reserve the performance of many of our most important rituals and visible positions of authority, respect, and honor for men.
Yeah, great questions. This gets confusing because priesthood can mean very different things depending on our definition. 1) is the authority to lead, administer, make decisions, and sign checks in the church. 2) is the authority to perform community rituals. In the kingdom of Israel under the law of Moses the king was anointed to be in charge and the prophet and priests had purview over the rituals. Our church tends to combine these roles under the same umbrella term of priesthood. It is these roles that require priesthood ordination, male genitalia, and for over 100 years the absence of African ancestry was also required. 3) Then there is the broader priesthood definition of the power of God delegated to mankind generally and that does not require priesthood ordination, membership in our church, or even really a belief in God to access. Women have that last definition of priesthood power (and also the power to perform specific delegated tasks relevant to her church assignments).