Great question, @Jason.Paul!
I would say that it depends on how you define the word “church”.
There are some who say that the Greek word ekklesia simply means a “called out assembly”. And so the church began when Christ first called out an assembly of 12 to follow Him during His earthly ministry.
But in that sense, Acts 7:38 calls the Jews in the wilderness under Moses a church. So that definition seems a little vague - a little too inclusive of things that we don’t really mean when we think of the New Testament body of Christ made up of believers who’ve all been united by the indwelling Spirit.
If you mean Church in the more NT sense, then you’ll need the indwelling Spirit at Pentecost to unite the believers in one body.
The Bible usually uses the word church in a local sense to refer to a specific body of believers who meet together - the church at Corinth or the church at Rome, for example. This is often referred to as a “local church”. It is made up of pastors, saints, and usually deacons (Philippians 1:1).
But sometimes the word Church refers to something larger - all the saints around the world and across the generations who make up what is often called the Universal Church. This was the sense that Jesus used the word when He said that the gates of hell would not prevail against the Church in Matthew 16:18.
As to why the church was begun, Acts 2:42 appears to address that issue.
I hope this will help you with this question.