Personally, I think that the guy who gave away the hot tub knew that he had a bad heating element. If that's not it, the tub having been drained and moved could have caused any potential heating element issue to hurry itself along.
I like the idea of disconnecting it and showing this guy that it cycles fine with the heating element disconnected. Doh! I sat here thinking of all kinds of "techno" ways to show the guy that it is the element, but in reality, the simplest way would be more obvious.
I don't think that I've ever seen this problem where it wasn't the element anyway.
Thanks very much to everyone for the ideas so far.