In the early church there was no such thing as a full time paid position.
There were missionaries, like Paul, who planted churches. But as important as their work was, they did not receive a regular payment from the church, but instead worked part time to support themselves. It is true that they sometimes received gifts to help them continue in the Lord's work, but that is a very different thing to a paid position.
When a new church was established the missionary would appoint elders which included deacons and bishops (plural). But once they were appointed the church was self-governing and were therefore responsible for appointing their own leaders in the future (unfortunately they did not always choose to be guided by the Holy Spirit).
It is a common misunderstanding in the church today that each of the early churches had a single overseer (bishop); but that was not the case. See Were Early Churches Ruled by Elders or a Single Bishop? by Michael Kruger.
In the early Church each church was run by locals. Each church was run by a group of elders who worked part time. They did not build for themselves earthly kingdoms; instead they preached the good news about the Kingdom of God and served the local congregation and the community.
There was no such thing as denominations; there was no such thing as Baptists or Presbyterians. Each church was known by it's location (e.g. The church of Corinth).
Why is this important? Because centralisation, by its very nature, leads to unnecessary problems.
Now you might say, "Our church is governed by a group of elders. The elders appoint the pastor." But what the person who says that usually means is: a man or woman who has passed their exams at Bible College (almost always an outsider), gets to preach at their church "X" number of times, and then the elders decide whether that person is going to be their pastor.
The process of choosing a pastor the way most churches presently do, is fraught with danger for the congregations that choose them. ( As we are all aware, there are sexual predators who seek positions of power.) Sometimes churches get "stuck" with a pastor who has real problems; but they can't find someone to replace him so they put up with it (see here). In the early church they never had this problem because they were never dependant on one person. It was much easier for elders in the early church to remove someone from a position of power.
Now just to be clear, many pastors are good people (as far as people go), but it is also true that many of them are overwhelmed with the burdens placed on them. (Which would rarely happen if the church structure outlined in the Bible was adopted.)
Now it is a good idea to have one or more elders trained in things such as funerals and marriages, but the elders could choose from their own members as to who could perform those roles.
Some will say an overarching authority is needed to ensure doctrinal purity. But an outside governing body is not necessary to ensure that. The elders, in their church, if they so chose, could make a rule such as "If you wish to be an elder in this church, you must have been a member of the congregation for X number of years and you must have attended XYZ Bible college for a year."
And that's one of the beauties about a self-governing church who is not bound to a denomination. If XYZ Bible college starts teaching wacky things, the elders of that church can discuss it, and if they so decide they can say, "from now on if you wish to be an elder in this church, you have to attend QRS Bible college for a year."
Would there still be parachurch organisations? Of course! And under the structure proposed, individual churches would continue to be enriched by them.