The common simile of a chariot actually does a poor job of getting this across, since you can take apart a chariot into wheels, etc. The apple simile is better, because you can't take an apple apart into "redness", "sweetness", and "softness".
So, you can analyse and action or an emotion into a "physical" and "mental" part, but you can't separate them. Hence this whole question of "is dukkha just psychological?" is problematical. In Buddhism there is no clean body + mind split. (And, it seems, not in modern psychology either, but I'm no expert on that).
I hope this is the point that you are trying to get across. Please correct me if I'm wrong!
In all fairness, there are a few things that make this harder to understand.
(1) Its more complex than this. You really can separate out the 'redness' from the 'sweetness'... someone who is colorblind does this. For someone who is not, under certain lighting, the light hitting the eye will no longer be red. This is where it gets more complex because that person will almost certainly see red even though the light does not suggest red. This idea of color constancy is one heuristic we employ to maintain a coherent consciousness, and it is a great example about how things get really confusing. Whose in charge? What if you see a yellow apple? or a Golden apple? Its not so easy.
This is why I usually take a different tack and talk about how the body actually does quite a lot of the processing. The body is part of the mind, and if you lose that part of the body, you lose the mental capability represented in that part of the mind. They are not separate at all. Any argument you can make to suggest that the body is different than the brain you can make about parts of the brain as well.
(2) The age of a person goes a long way as to determining how entrenched the philosophy of Dualism is. Just because younger generation med students (who have already had lots of coursework leading up to med school, and generally share certain characteristics) get it easily does not mean that someone coming from a different background, with different ideas, will get it easily. It also means that sometimes a different approach is necessary! :p