Were the three Wise men representative of each of the three human races?

No.

The Bible doesn't mention how many "wise men" ["magos" - Magians, Oriental scientists or magicians - Strong's Definitions] traveled from Babylon to Jerusalem to investigate the "signs of the times" that they recognized from the prophecies in the Scriptures as many as "two years" prior to their arrival at the door of "the house" where they finally stood face-to-face with [or bowed down to] the "young child" - possibly closing in on two years of age, "King Jesus" [Matt.2:11]

Halley's Bible Handbook reveals:

"These Wise-Men came from Babylon, or the country beyond... the land of Abraham, land of the Jewish Captivity, where many Jews still lived. They belonged to the learned class, advisers of kings. Perhaps they were familiar with the Jewish Scriptures, and knew of the expectation of a coming Messianic King. It was the land of Daniel, and no doubt they knew of Daniel's 70 weeks' prophecy... They were men of high standing, for they had access to Herod... But the Scripture does not say how many. There were probably more, or at least they were with an entourage of scores or hundreds, for it would not be safe for a small group to travel a thousand miles over desert wastes that were infested with bandits..." (Halley's Bible Handbook with the King James Version; Classic Edition: p.418 Chap.2;1:12 Visit of the Wise-Men)

The modern "holiday fables" passed down to our generation are not in accordance with the Truth of the biblical account; where the "wise-men" never came to the manger, but were led by the 'miraculously moving star' [an angel - see Rev.1:20: {"...the seven stars are the angels of the seven churches..."}] that "...went before them, till it came and STOOD over where the young child was..." - Matt.2:9 as many as two years later.

The number of the "wise-men" in the biblical account can't be representative of the three [main?] races of mankind, because the Bible cites no such number.