Menachem, there is no Mesorah specifically to shave. For a long time, there was no halachic way to fully shave. There is a Yerushalmi that mentions that is was common to trim beards using scissors.
Jews in Europe generally followed the customs of the country they were in. When the custom was to fully shave, they used chemical depilatories for it. When electric shavers were introduced in the 1900s, it presented a question of whether they are halachically allowed or not. R’ Moshe ruled that they are permissible.
That said, it was not accepted by everyone that fully removing your beard is allowed. The Tzemach Tzedek held that it violates the issur of a man appearing as a woman. The Shulchan Aruch disagreed, and held that this issur relating to hair meant that a man shouldn’t shave hair from body parts that women usually do, such as underarms.
Regarding this last point, I will note that I once had a conversation with R’ Bender about the issur of Lo Silbash. I questioned whether it changes with the times and societal norms – for example, nowadays, it is common for women to wear pants (the issur isn’t limited to what frum Jewish women wear!). So would it still be assur for a woman to wear pants due to this issue? He told me that no, it is not considered a man’s garment anymore. He said the first women to wear pants, if they were Jewish, did violate the issur, but once it became common for women to wear pants, it’s not prohibited due to this issue. The issue of tznius is a different problem, which he said is a major issue with pants. However, he said that if there were pants that were completely tznius, it would not be a problem for a woman to wear them.
I’d suggest that the same reasoning applies to shaving, and even according to the Shulchan Aruch’s view. Nowadays it’s completely normal for men to shave their beards completely, and even to remove hair from other body parts. Since this is the societal norm now, it’s possible that the issur of appearing like a woman no longer applies to this area.