Who Can Baptize? Part 3

Kevin DeYoung's post on who is authorized to baptize had four arguments. I've discussed the first two already, the biblical and theological arguments.
The third argument is "exegetical": "an appeal to the priesthood of all believers does not support the administration of baptism by every church member. ... It does not suggest that now in the New Testament there are no rites which may be performed only by ordained officers."
My response is that he makes a great point that the argument from the priesthood of all believers, by itself, is not convincing. If that was the only argument for allowing non-pastor's to baptize, then it would be a fairly weak biblical case to make.
The fourth argument is pragmatic: "for baptism to be responsible there must be some church oversight. ... There must be a process of accountability and evaluation. Invariably, as Grudem points out, the pastor(s) of the church are likely involved in determining who can be baptized and who can baptize. If church officers superintend the process ... it stands to reason that they exercise their Christ-given authority in performed (sic) the baptism itself."
This argument, which, to be fair, he himself considers his weakest, is puzzling. Why can't "church officers" superintend the process and delegate the actual baptizing to a member in the church? Why can't "church officers" evaluate the candidates, but leave it up to other to perform the actual baptism? He starts by saying "there must be some church oversight," but concludes by implying that the church leadership must perform all of it. We can have church oversight while a lay person does the actual baptizing.
Granted, I do appreciate the concept of church oversight regarding baptism. I think it is very wise. I'm not sure it's required, but it's wise.
These are my thoughts on non-church leaders performing baptisms. I have no problem with it. I think Scripture never intentionally addresses the issue, with the closest being Acts 9:18 where Ananias baptizes Saul/Paul.