Restore air traffic controllers' instructions from aircraft tracks
No - there are just simply far too many permeation's of instructions. 

For example - an aircraft climbs 1000ft. This could be because:


It has just received an instruction to do so
It previously received an instruction to do so at a certain point
It is operating within a block clearance and chose to do so
It experienced an emergency or TCAS resolution instruction and did so without instruction
It made a mistake/pilot deviation and incorrectly did so


And that's just one minor change, with one aircraft. You can certainly take a guess on some of them but even ignoring unusual situations the first three possibilities are common and normal practice.

And that's just traffic presumed to be operating IFR within controlled airspace - add in VFR traffic in uncontrolled airspace (And bear in mind that they can be still be receiving instructions and clearances at that point - for example, entry into controlled airspace or landing at an airport in uncontrolled airspace) and you've got, essentially, no chance.