Are there theological explanations for why God chose the Church to preach the gospel instead of more effective and efficient alternatives?

What if the end goal is not "the preaching of the gospel to every human on the planet" as your question stated? What if the end goal is humanity transformed and that the preaching of the gospel is a means to this end?

God is not known to be efficient, but He gets the job done (humanity transformed) by eventually bringing all whom He has chosen to His sheepfold before the night comes. My answer starts with several Biblical observations and theological reflections and concludes with an answer from a reputable theologian (Michael J. Gorman) framing mission as PARTICIPATION.

Biblical observations and theological reflections

• Adam caused the Fall and through Adam and a line of sinful people that the world will be saved. Note how in the genealogy of Jesus (Matt 1:1-17) we have Jacob the liar, Tamar who slept with her father in law (Judah), Rahab the prostitute, and Bathsheba (whom God chose out of David's other wives, even though David murdered her first husband Uriah!)

• It's a Biblical pattern that God chose people with problems BUT who have hearts for Him. He dignifies them to be involved in His plan. He is acting like a leader in a company who wants to include everyone in the company's success.

• He even helps them to overcome their weakness (like Moses and his speech issue, and Gideon with his fear). He is acting like a father who loves to involve his little children who offer to help their dad to build / repair something although he would have finished the job sooner by himself.

• In Christian teaching, a person cannot come to faith unless God first choose him/her (doctrine of election) and then provides him/her with grace to respond to the gospel. Those two are essential, but miracles and vision are not. At the end of the day, despite our evangelizing efforts and powers of persuasion, it is still up to God and up to the person we are evangelizing to.

• One reason why God works through His Church is that He wants us (the problem) to be part of the solution. God is control but wants us to be maximally involved. In doing so we become more aware of our sins but become better as a result of evangelizing. His providence and foreknowledge make it work in the end. We look forward to becoming amazed at the extent of God's wisdom when everything is revealed at the end of time.

• Paul is our model as his own life becomes the gospel: he started by being a murderer of God's elects, repented, and voluntarily became a soldier for Christ, suffering for Him out of his gratitude for being saved, and wanted everyone to have the same experience.

Michael Gorman's book: Becoming the Gospel

For a reputable theologian's answer, I refer you to the 2015 book by NT scholar Michael J. Gorman : Becoming the Gospel: Paul, Participation, and Mission reviewed here with some constructive criticism here.

Among the points in the final chapter of the book there is one theological explanation which I paraphrase as follows: God's mission is the formation of a new humanity, so the gospel has to be preached out of a bodily witness of the transformation.

Relevant quotes from the chapter (emphasis mine):

The Gospel and the Missio Dei

A critical part of becoming the gospel is appropriately defining the gospel. A narrow, privatistic understanding of the gospel (such as, “Believe in Jesus as Savior and Lord so that you will have your sins forgiven and go to be with him when you die”) may require the church to say a lot, but it will not require it to become much of anything other than a loud voice. It does not even have to be a very credible loud voice as long as it says what (it thinks) needs to be said.

...

This is why our understanding of the gospel and our understanding of the missio Dei are so interconnected. A thin, lightweight view of the gospel entails a similar thin, lightweight understanding of salvation and the missio Dei. But a thick, robust understanding of the gospel involves an equally complex and comprehensive perspective on salvation and the missio Dei. What God was and is up to in Christ is ultimately cosmic in scope, but in the present that future cosmic reality is anticipated in the formation of what Ephesians refers to as a new humanity. If the gospel of Christ crucified, raised, and exalted as Lord means anything for the human race, according to Paul, it means that transformation into the cruciform image of God is possible in the present. And that means that the message and the people, the gospel and the church, are inseparable; the witness to the reality of transformation through death and resurrection is the existence of a transformed and being-­transformed people.