I would have no problems at all with CC shadowing a flight crew for the day. An understanding of some of the things involved in the job and the workload and considerations is positive, and will hopefully lead to a better understanding to why, for example, sometimes we are too busy and why it will take some time to respond to them.
I would stop it at that though, I would not tell them how to operate any flight deck switches (other than possibly the r/t ptt switch for use in speaking to Medlink) and I certainly wouldn't want them near any flight critical items.