By the time the Magi reached Jerusalem it was much later. Perhaps two years [Matthew 2:7] based upon the information Herod received [Matthew 2:16].
The custom of giving gifts to a King....or Head of State was established protocol long before the birth of Our Saviour. The fact that the Magi presented gifts to Him had absolutely nothing to do with His being born.....but the fact He would be King....according to prophecy.
THAT;S why we give Presents....to HONOR the RECIPIENT!! Lights for the STAR, etc....it’s NOT PAGAN, bro.
Gift giving during the Saturnalia observance had been customary long before the birth of Our Lord.
I should have read a little further before I responded...:-)