This could be done and is actually being discussed in some circles. Some practical problems for this relate to bandwidth requirements to transfer data and onboard recording/processing devices to package and send the data. Many data points bubble up and traverse the ARINC 537/717/429/575 data bus at a fairly high rate of occurrence, around 1/6th of a second for every input (dozens on newer aircraft). Couple that data with audio, and possibly video (if Mary “I don’t understand economics” Schiavo gets her way), and you have a large data package to squeeze into a narrow tube (VHF). While there have been advances in VHF data links, you are now talking about nearly continuous data streams clogging the airwaves.
I am a proponent of information transfer, not data connections for the sake of data. If you have smart devices (data concentrators) onboard the aircraft to recognize significant deviations from the standard flight profile, have triggering events to turn it on, you could establish a situation where data/information are jettisoned - almost like jettisoning the Captains log.