NC: You misunderstood me. The idea that the current Minhag of some to wash before Kiddush is related to Shittas Beis Shammai in the Gemara in Berachos is a misunderstanding of that Gemara. I apologize if I wasn’t clear.
1.4: No proof. Nor any real way to prove it. It’s not that there is a third Gemara. It’s that there was a third major Yeshiva that is more Yerushalmi-based than Bavli-based. It has no proof. It’s just a theory that answers a lot of questions.