We covered why we sing in worship a while back, but who sings is a different question. I actually could not find an answer about why lay people, either someone assisting with worship or a designated singer, also called a cantor, almost always lead the kyrie. My gut says it is simply a part of worship that does not need to be led by a pastor so a lay person usually does it. More theologically, it reminds us that the cries of “lord have mercy” belong to all the people. So anyone, regardless of whether sung or spoken, assisting with all or just that part of worship, represents the cries of God’s people.