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Am I advocating taking three-month old believers and giving them oversight over churches (the very thing that Paul did)? Yes, but only if those believers meet the biblical requirements for elders/overseers, and only if they are given oversight of churches that follow a biblical model. That is, those churches must first of all be newly-planted gatherings that are submitted to a mature founding minister, such as an apostle, who can provide some oversight. That way, newly-appointed elders are not entirely on their own.
Second, the congregations must be small enough to meet in homes, as did the early churches. That makes churches much more manageable. That is probably why one of the requirements for elders/overseers is that they successfully manage their own households (see 1 Tim. 3:4-5). Managing a small "household of faith" is not much more challenging than managing a family.
Third, the congregation must consist of people who have responded in repentance to a biblical gospel, and who are thus genuine disciples of the Lord Jesus Christ. That eliminates all of the challenges that arise from trying to pastor sheep who are actually goats.
And fourth, the pastors/elders/overseers must follow their biblical role rather than a cultural role. That is, they must not hold a central, all-important, spotlight position as they do in most modern churches. Rather, they must be single parts of the entire body, humble servants who teach by example and precept, and whose goal is to make disciples, not by being Sunday-morning orators, but by following Jesus' methods.
When that pattern is followed, then some three-month-old believers can oversee churches.
 In Paul's first letter to Timothy and his letter to Titus he mentions leaving them behind to appoint elders/overseers in the churches. So Timothy and Titus would have provided oversight to those elders/overseers for some time. They would have probably periodically met with the elders/overseers to disciple them, as Paul wrote, "The things which you have heard from me in the presence of many witnesses, these entrust to faithful men, who will be able to teach others also" (2 Tim. 2:2).
 See Acts 2:2, 46; 5:42; 8:3; 12:12; 16:40: 20:20; Rom. 16:5: 1 Cor. 16:19; Col. 4:15; Philem. 1:2; 2 John 1:10.
 It is notable that Paul's letters to the churches are addressed to everyone in the various churches, and not to the elders or overseers. In only two of his letters to the churches does Paul even mention elders/pastors/overseers. In one instance they are included in the salutation, added as if he didn't want them to think that they were excluded recipients (see Phil. 1:1). In another instance Paul mentions pastors among a list of ministers who equip the saints (see Eph. 4:11-12). It is also especially notable how Paul makes no mention of the role of elders as he gives certain instructions that we would think would involve elders, such as administrating the Lord's Supper, and the resolution of conflicts between Christians. All of this points to the fact that elders/pastors did not hold the central, all-important role that they hold in most modern churches.