There is constant data sent out by most aircraft already. The black box is for when shit goes very wrong, like when a plane is on fire, out of power, with a big hole in it, falling like a rock and spinning like a top.
Addition: the NTSB is expert at retrieving data from severely damaged black boxes.
Redundancy is good, always. Especially when the cost is negligible compared to the lifetime operating expense of the machine and the costs from any misinformation regarding the last moments of a doomed flight (a Crash that is the Airline's fault mega-screws them in the wallet).
Having said that, your question does resonate well with me too. How much data does a black box store that we can't send it over satellite (or even to other aircraft nearby-- think of a distributed storage) in real-time or near real-time? The purpose of such distribution is purely redundancy. I haven't seen any concrete numbers or engineering barriers against this idea, yet.
There's also a practicality argument: when a plane is flying along normally, you don't need consistent satellite information - you've got radio communication, radio data-links, and controllers watching. So the usefulness of a satellite uplink only comes when something is not right with the plane.
If we're in that situation - that there's something wrong enough with the plane that you'd want to know about it via the uplink - what makes you think the uplink will still be working? Air planes are designed with huge levels of redundancy and overengineered margins: are you going to take all that, double it, and then put it into a satellite transmitter (which probably won't have that much data to send anyways if the plane is in that state)?
So no, I don't think we need constant satellite uplinks. Black boxes (especially ones of the newer generation) are very well-engineered, and the NTSB guys are masters at getting things off broken ones.