Please delete my answer.

I suspect the answers is two fold. To be useful you would need either continuous communications or at least the reliable ability to communicate from any point on the globe even after a crash. Firstly therefore, you would need a global network of ground (and more problematically sea) stations to receive the data. You could of course use satellites but either option would be expensive and require international agreement. Secondly, the existing standards were probably drawn up before the days of widespread digital comunications would have made it techincally feasible, even if you could afford it.

Air disasters of the scale where black box recovery proved problematic are rare events in percentage terms. As per the previos answer beyond logistical challenges of networking this information realtime, the commercial cost of such an initiative is better invested in ensuring planes stay in the air.

Various teams are already working on that idea, but it is not simple. To design and support reliable means for capturing all the data from all the commercial flights in the air at the same time is demanding, and commonly the need is greatest just when it is hardest to ensure communication. Flight data recording on-board as well as remotely would be ideal, but costly. It would be nice if live overseers back on the ground could periodically read the recordings in real time and ask what the pilots thought they were up to!

But the sheer volume of communication and data processing would be demanding. Even setting data standards would be challenging; each model of aircraft produces different data; flight data recorders don’t just record cockpit chatter. However glibly we speak of “black box recorders”, they differ. Some older models barely recorded more than the most recent half-hour of cockpit conversation, plus a few equipment states. Such scanty records have frustrated several accident investigations in the past, even when the data was recovered successfully. The current tendency therefore is to demand more channels of data than generally were recorded in the past. Black boxes are attractive because they seldom are needed (fortunately) so before a crash no one complains about their reliability, and some airlines skimp on their options.

There are at least two main objections to doing this:

The first is cost, as it involves using relatively expensive satellite bandwidth to transmit to off-plane storage.

The second is human-related: Pilots have raised objections to being kept tabs on in this manner. Bear in mind that there are two "black boxes", one to monitor flight data, the other to monitor cockpit voice conversations, often critical in indicating what went wrong on a flight.

Some have raised the possibility of a "panic button": Normally the on-board black boxes would make the only recordings, but in a seriously abnormal situation, pilots would have the option to enable live-transmission via satellite to off-plane storage. Or perhaps the enabling could be done automatically when flight data exceeded aircraft design parameters (e.g. rate or angle of descent) for more than a certain interval. To my mind the "panic button" provides a happy medium, albeit one dependent on human judgment and rationality.