To test for general intelligence, you can't test on the specific skill the bot's trained in.

I think we might have different definitions of what "general intelligence" is. I thought it meant something like, "being able to solve novel problems in some domain"; in this case, our domain is "human conversation". I may be willing to extend the definition to say, "...and also possessing the capacity to learn how to solve problems in some number of other domains".

Your definition, though, seems to involve solving problems in any domain. I think this definition is too broad. No human is capable of doing everything; and most humans are only good at a small number of things. An average mathematician can't compose music. An average musician can't do calculus. Some musicians can learn calculus (given enough time and motivation), but others cannot. Some mathematicians can learn to paint; others cannot.

Perhaps you mean to say that humans are not generally intelligent, and neither are AIs who pass the Turing Test ? In this case, I might agree with you.