I’ve got one insight that I wanted to share because I’ve been wondering about it for years!

Why were just Peter, James, and John invited to the Mount of Transfiguration? 

According to Joseph Fielding Smith they were the first presidency of the New Testament church after Jesus’ death (source). These three men had also been the only witnesses to Jesus raising Jairus’ daughter from the dead (the first time He had performed such a miracle) and were later the only ones invited to watch and wait with Jesus in Gethsemane. 

Jesus seemed to be preparing Peter, James, and John to lead the Church and serve as special witnesses after His death. (Source)