I think they do to some extent... the so-called ACARS data. Not sure why they don’t do that with the voice data too... maybe it’s out of range in certain areas?
ACARS is the company phone...not much else.
It’s a bandwidth limitation. ACARS uses a high frequency radio to communicate, which is ‘slow’ to transmit data but has very good range. Basically, if you streamed the avionics telemetry from a plane in real time, you would exceed the bandwidth capability of an empty cell phone tower - per plane. In the case of cockpit audio, there’s also the issue of privacy and confidentiality.