Again taking this seriously as if it would enhance safety...
You could have certain types of faults and abnormal conditions trigger 'immediate mode' which would immediately upload the last minute or two of FDR data. It is what happened just before the abnormal condition that is important.
The problem with any approach is that by definition such a reporting system would be nonessential. How much power do you want to use from a dying battery to send a message back to base, when modern aircraft do not stay in the air without some sort of electrical power? It is very hard to engineer something that can't in some rare case either cause a problem or make an existing problem worse. The safety of living passengers far outweighs the curiosity of the rest of us.