This is an area that greatly interests me, so I thought I would have a shot at it..
Firstly, Priesthood in the Old Testament came with certain advantages and certain disadvantages. The Sons of Aaron and their Levite helpers were given no inheritance, so they relied on Tithes for theirs and their families upkeep. They were a paid ministry.
It seems to me, that it quickly got messy. Who had a legitimate claim to the functions of an Aaronic and Levite Priest and who didn't? I think there were a lot of claims and 'endless' genealogies to try and prove a claim.
The roles of the Levitical and Aaronic Priests were quite specific, particularly in terms of their roles within the temple, which was mainly based around sacrifice and certain functions during various religious events during the year.
This isn't how the Aaronic Priesthood functions in the LDS church today. Or so it seems to me. There may be some superficial similarities but that is where it ends.
Secondly, Jewish Society wasn't one cohesive whole. There were various movements with various beliefs, some more patriarchal than others.
Thirdly, When Jesus was told by the Apostles that others were fulfilling duties that he had specifically set for them, he is portrayed as having a liberal attitude in terms of the authority to carry out work in his name as seen in Mark 9
John said to Him, “Teacher, we saw someone casting out demons in Your name, and we tried to prevent him because he was not following us.”But Jesus said, “Do not hinder him, for there is no one who will perform a miracle in My name, and be able soon afterward to speak evil of Me.“For he who is not against us is for us.“For whoever gives you a cup of water to drink because of your name as followers of Christ, truly I say to you, he will not lose his reward.
Actually, the Didache supports this idea. Written at a time when there were still travelling Apostles, Prophets, Missionaries and Teachers. The guidance given was that insofar as they taught in accordance with the Two Ways, they should be accepted, but if they started to take advantage or stay longer than a few days they were false Prophets/Apostles. Full time teachers that were of benefit to the community (this particular one anyway) were worthy of being paid.
Paul does mention a variety of women in his genuine letters, many of whom held positions of authority whether that be leading house churches, financing ministries, and teaching. The Eucharist in the Didache was more of a Meal, rather than the Sacrament that is conducted in the LDS faith. Interestingly, Jewish influence is indicated by the drinking of wine first. I'm not saying it matters, I don't think it does. But things have changed and developed in terms of praxis and doxy over time.
I guess what I am getting at, is that it seems to me that Priesthood wasn't necessarily required for many of the ordinances and leadership within the early church. That fluidity didn't last long, because, well most organisations tend to slide towards orthodoxy and institutionalisation. In a way, they have to, to survive.
It may be that Jesus attitude towards women, who followed him, whom he taught (Mary sitting at his feet), was just too anarchic for wider Greek and Roman Society. Philo and Josephus were quite dismissive of women on the whole. I think a good argument can be made that Paul's comments in 1 Corinthians were a later interpolation, and that initially at least, when the church was eschatological in outlook, women met men on an equal par. As Christ's return was seen to be delayed then in order to survive, the orthodox movements moved towards a Patriarchy that we still feel the effect of today.
However, as the Montanists and various Gnostic Groups indicate, there were movements that survived for a while that were more egalitarian in their approach to women.
"It's a little like the Confederate Constitution guaranteeing the freedom to own slaves. Irony doesn't exist for bigots or fanatics." Maksutov