Im sure this happened widely and not isolated in egypt it would make alot of sense.
It makes sense to our modern perspective, but it may not have in the classical world. The Romans and Greeks did not have our concept of professionals. An academy would be visited by gentlemen of leisure rather than trainees hoping to get a job. I may be wrong, but you'll have to present some evidence that it was academies rather than on-the-job training that produced scribes and healers.

The system may have existed in Egypt and the other bureaucracies of the east, but these were now ruled by Hellenes.

As for nomadic faction. and academie is a long shot. but a wise ruler would definetely get a wise shamen scribe etc to teach his kids siblings. like genghis khan did to his siblings and kids
Again, obviously the nomads would have had some form of education. But that's not the same as an academy.