What tradition teaches and what is truth varies considerable especially throughout the gospels.
Tradition teaches three wise guys…. because of the three gifts. But they traveled in caravans (not minivans) more likely around 12 or more peeps
The gospels were written from 4 different perspectives (see Below)
Scripture build-up regarding a particular event can be gleaned from each gospel. Some as you noted may have nothing; others may give a portion. By putting together scripture from each you will have what the Bible provides.
As for the birth…..angels and shepherds were the only ones there. The wise guys came less than 2 years later to the home.
The gospels were designed to show the four different aspects of Jesus Christ ministry ….as the prophets of old foretold them in regard to the BRANCH in the OT.
Matthew - as King.
Mark - as Servant.
Luke – as Man.
John – as Son of God.
Below gives the Scripture reference and more details. ….
KING
One quality of the coming BRANCH prophesied by Jeremiah would be that of a King descended from David.
In order for Jesus to be the Messiah He would have to be a descendant of David inherit the right to the throne and come as King to Israel.
Jeremiah
23:5 Behold, the days come, saith the LORD, that I will raise unto David a righteous Branch, and a King shall reign and prosper, and shall execute judgment and justice in the earth.
33:15 In those days, and at that time, will I cause the Branch of righteousness to grow up unto David; and he shall execute judgment and righteousness in the land.
Only in the book of Matthew do you find the phrase “Kingdom of Heaven” and it’s used 32 times.
There are 10 parables unique to Matthew which relate to a King. Plus the royal genealogy chap 1, and the phrase son of David occurs more in Matthew than any other gospel.
SERVANT
As the promised BRANCH one of the aspects of Jesus Christ ministry would be the role of a Servant.
Zachariah 3:8
3:8 Hear now, O Joshua the high priest, thou, and thy fellows that sit before thee: for they are men wondered at: for, behold, I will bring forth my servant the BRANCH.
The gospel of Mark emphasizes Jesus as a Servant. There is no genealogy as a position gained by descendant. The word “lord” or “sir” Greek ( kurios) is used only 3 times of Jesus in Mark….. whereas it is used 73 times in the other three gospels.
There is a great emphasis in Mark on Jesus’s actions and responsibility in serving and helping others.
MAN
Another characteristic of the promised BRANCH is Jesus’s humanity …in other words, the Branch would be a man.
Zechariah 6:12
6:12 And speak unto him, saying, Thus speaketh the LORD of hosts, saying, Behold the man whose name is The BRANCH; and he shall grow up out of his place, and he shall build the temple of the LORD:
Luke clearly emphasizes Jesus Christ as a man. Luke 3:23-38 is the genealogy tracing Jesus’ legal ancestry …through Joseph (who adopted him) back to Adam the first man.
There are 11 parables particular to Luke which emphasize the human aspect of his walk. The entire gospel emphasizes His relationship with common man, including publicans and sinners…..It shows Jesus to be a full of human tenderness and compassion as the perfect man.
SON OF GOD
Isaiah 4:2
4:2 In that day shall the branch of the LORD be beautiful and glorious, and the fruit of the earth shall be excellent and comely for them that are escaped of Israel.
Jesus Christ was the Branch …the offspring of the Lord God. The gospel which clearly depicts and emphasizes Jesus as the son of God is John as seen in the following verses.
John 1:14, 18 & 34 - 3:16 & 20:31
1:14 And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.
1:18 No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.
1:34 And I saw, and bare record that this is the Son of God.
3:16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
20:31 But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.
While every gospel encompasses all of these characteristics of Jesus Christ, each gospel specifically emphasizes one of them.