The “source” is from the Torah (Bereishis 24:65), where Rivka Imanu covered her face when meeting Yitzchok Avinu. It is also brought down in the Gemorah (Ketubos 17b, Rashi ad loc).
This is incorrect. To see this, note that nowadays the choson bedeckts the kallah. However, as you wrote, Rivka Imanu was bedeckt herself. So if this were true, as most people beleive, then the kallah should bedeckt herself.
Neither the gemorah nor Rashi say that the choson bedeckts the kallah. All that is said is that the kallah wears a veil.
As you noted, the Tur and Tosefos hold that today’s bedecken would actually be the chupah. The Bach does state the we try to be yotzei all shittos, but note that bedecken today wouldn’t be yotzei the Tur or Tosefos’ shitah.
To see this, note that nisuin can take place only after kiddushin. Since at the time of bedecken, kiddushin has not yet taken place, it would be an invalid nisuin.
I apologize, however, in that I should have been more clear. The gemorah IS the source for wearing a veil, and you correctly stated the opinions of the rishonim. What I was trying to say was that the custom of bedecken as it is performed today is baseless. It couldn’t accomplish nisuin, so it is ineffective.
I retract. Apparently Rashi proved that nisuin can precede kiddushin. He brings the following case: The arranged marriage of an orphaned ketana by her mother must be repeated after she is a gedola, but her chuppah need not be repeated. It’s like people say: you learn something everyday or you don’t. I had been told by a former maggid shiur that bedecken was pointless, but I can no longer find a limud z’chus to that in light of this teshuva from Rashi (I believe it’s from a Hagahot Mordechai that was maybe quoted in the Kehillas Yaakov)