The Great Commission, as it is now called, was recorded in Luke's second reference volume we call The Acts of the Apostles.

The nature of his work is written in the first chapter of his first volume. "Many have undertaken to draw up an account of the things that have been fulfilled among us..." That begins to explain that he was not an eyewitness, but researched the accounts of others who did.

By the by, , as a physician, Luke would know how to do research and diagnosis to develop his Gospel.

As for the three other Gospels, I acknowledge that there are different opinions, but to avoid am attempt at a lengthy dissertation, allow me to refer you to this website: https://www.blueletterbible.org/Comm/stewart_don/faq/historical-accuracy-of-the-bible/question10-when-were-the-gospels-written.cfm

As far as the virgin birth, Professor Emeritus John Lennox of Oxford probably talks about it more thoroughly, but as for us, because we know God, the Creator of everything, really does exist, the idea is not at all difficult for us to accept.

As for its necessity, our inner person has a bent for doing hurtful and harmful things - being born of the flesh - while the Spirit of God does not, and therefore, Jesus conception did not leave that bent in Him.

The unfolding of the panorama of the Gospel rolls open from there.

Of course, if you don't know God, Jesus, or the Bible, anyone will naturally have doubts. So, I do understand your unbelief.

My wife and I were not always Believers. However, we are now Born Again Bible Believing Christians... a necessity, Jesus pointed out, for people to see the Kingdom of God.