It is irrelevant that rich travellers would arrive when He was first born.More relevant it was for the poor and outcast like the shepherds. But it is relevant that they brought gifts that would help pay for an escape to Egypt. As a matter of fact it doesn't even say three.of them. They saw the star in the east yet they were travelling west so that means they first saw His star, no doubt Jupiter, the king maker, when they were in the east themselves. Also a new chapter indicates a different time, as does young child which does not correlate with newborn.
Regardless, it says... Matthew 2:9 When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was. 10 When they saw the star, they rejoiced with exceeding great joy. 11 And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense and myrrh. 12 And being warned of God in a dream that they should not return to Herod, they departed into their own country another way.
A Christmas tradition has overtaken scripture much in the same way other traditions within the religion have, especially a tradition that those of
God can align with governance of man when the original scriptural purpose was to remain loyal only to the will of God, creating a division, between world of man and Kingdom.