For every high priest chosen from among men is appointed to act on behalf of men in relation to God, to offer gifts and sacrifices for sins. He can deal gently with the ignorant and wayward, since he himself is beset with weakness. Because of this he is obligated to offer sacrifice for his own sins just as he does for those of the people. And no one takes this honor for himself, but only when called by God, just as Aaron was. So also Christ did not exalt himself to be made a high priest, but was appointed by him who said to him, "You are my Son, today I have begotten you"; as he says also in another place, "You are a priest forever, after the order of Melchizedek." (Hebrews 5:1-6)
Why don't we have a distinct priestly caste among God's people today?
It is so important to get this. I firmly believe that too many sheep have forgotten that they are just sheep. None of us are called to be above the church, to rule over other sheep. We are all sheep and we have one Shepherd. Many of us, although not nearly enough of us see it, are called to lead by example, to be be encouragers and equippers as sheep who have been following the Good Shepherd longer than newer sheep.
With Jesus Christ as our perfect, flawless High Priest we have no need for human intercessors or priests. Indeed we all minister in different ways as priests, not to offer sacrifices but to serve our God as His ambassadors, doing works of mercy in His name and proclaiming His Son to the lost.