mmm...interesting idea. I think I would like it, but it would have to be designed properly. Because training would have to take time, and if they are a high level trainer or something, there salary could be very high, and so by the end of the training the salary could just be ridiculously high and the training would have been a waste.
I think, rather than being able to train staff, I'd like to have a larger variety of staff available. So you still have Trainer, PR Manager and Doctor, but the skills of each available one might be different. So like you said above, Taping would be Strong etc, you could see a list of these skills for each Doctor on the market or something and each Doctor would be better at different skills.
However I think this would be very hard to code, and I would like to see a new draft system before this :)