Priesthood is not just authority to perform ordinances. It is more than just fellowship. These are both very important, but priesthood is also training in how to govern righteously, regardless of the setting, whether in families, church, or government.
The Aaronic Priesthood is the preparatory priesthood. It does perform certain ordinances and provides fellowship for the members of these quorums. But perhaps its most important function is to prepare for actual governing. There are presidencies where, usually young, men get their first lessons in holding responsibilities, follow-through, and delegating, as well as sitting in council to accomplish good things.
As they progress, the Melchizedek is the next step, where training and increased responsibilities continue in the righteous use of authority within the family and the organization of the Church. This is not all however. It is training for governing righteously in all settings, including government. However the most important is within the family.
Within the quorums men learn in theory and practice how to purify themselves, overcoming temptations that lead to sorrow and corruption. If we value this priesthood and work within it, the Lord will transform us to become as He is. THIS is the greatest of ALL miracles!