Did the gospel as we know it begin when Jesus first came on the scene?

No. The Gospel as we know it did not begin as soon as Jesus came on the scene, because no one understood that He was going to die. Even Peter and John did not know that He must rise from the dead, so they could not have preached that Gospel during His earthly ministry.

Les explains that the idea the Gospel as we know it began as soon as Jesus came on the scene is impossible, because no one understood that He was going to die. Peter and John, the men closest to Him, did not know the Scripture that He must rise again from the dead. If they had no idea of His crucifixion and resurrection, they could not have been preaching our Gospel during His earthly ministry. The Old Testament prophets had written of His death, burial, resurrection, ascension, and coming again—some count 360-some distinct prophecies—but it was in veiled language, and even the prophets themselves could not understand it. So these disciples had no comprehension of His death or resurrection. Les then moves to 1 Corinthians chapter 2.

John 20:9

For as yet they knew not the scripture, that he must rise again from the dead.

Could the disciples have been preaching the gospel during Jesus' earthly ministry if they didn't know He would die and rise from the dead?

No. Since Peter and John did not know the Scripture that He must rise from the dead, and no one understood He would be crucified, the disciples could not have been preaching the Gospel as we know it. That message of His death, burial, and resurrection was not yet understood by them.