If a tradition is what's handed down from one generation to another, than the revival must be part of the tradition.

This is just so not true. Revival and Tradition are chalk & cheese and it pays, I feel, to be aware of this as conflating the two just leads to all sorts of issues, none of which are helpful. The Tradition is largely a myth, an illusion consequent on Revival Methodology as what we're looking at are collected versions which form the fossil record of a long dead idiom of Popular Music Making. We're looking at that idiom, just so many shadows of it on the wall of Plato's Cave. For sure the two have overlapped - I'm honoured to have worked with Traditional Singers & Storytellers who have welcomed me with open arms, but never once was I under an illusions about my status as a Singer of Traditional Songs rather than a Traditional Singer. This isn't a matter of petty semantics, it's the very core of what Folk is.

That said there are (confusingly perhaps) revival traditions which are more orthodox conventions born of the inner religiosity of a movement adrift from mainstream culture by nature of its very convictions. But best we don't confuse them with anything truly Traditional, anymore than our Model Railway Enthusiasts confuse his authentic models with real trains.