They're already doing this, but there's no reason for it to be a random ID—It's an ephemeral ID assigned by the FAA. So you know your ephemeral ID & registration, the FAA knows your ephemeral ID & registration, and while everyone else can see your ephemeral ID they do not know the aircraft registration it's tied to.
An interesting image recognition project would be a camera watching departures from an airport that records the tail number of a plane and correlates it with the ephemeral ID to feed back into the exchange.
I'm not sure how well that would work w/r/t ATC hand-off and their ability to look up flight plans.
Furthermore, there's the problem of getting the random ID to ATC after the plane has "started up."