A churchwarden was/is a lay person in the Anglican curch that takes care of lay matters. So I guess it makes sense that the term is English and the pipe style smoked by some got that name.
Not sure that is correct. I'd go with it if the question had been had did these pipes get the name of "churchwarden". I believe these pipes originated elsewhere under a different name or names. My guess is Turkey.
Turkey, according to a Wikipedia article on it, although that doesn't make it true, IYKWIM.
To me, the idea that it owes its length to the long tavern clays has a certain appeal.