I should've also mentioned that in all my studies of ancient Egyptian religion, the professional literature I've read has never once mentioned "mystery schools" in any serious context. One might see them mentioned on fringe websites, fringe books, and by the fringy Rosicrucians, but that certainly does not make "mystery schools" a reality.
See, there was in fact a "mystery" school in the antiquity, but it had nothing, or very little, to do with what we would call "spiritual" secrets... the Pythagorean School had more to do with math...
But that would be many moons after the timeline we envision.
With all these "secrets" that are murmured we are talking in reality about "secrets of trade". All profession that required an extensive knowledge had its rites and initiations that sometimes go back a long time. First because certain actions and sentence forms help the memorizing, and second to keep unwanted competition away.
It is not surprising that the "spiritual" forms of initial took over the rituals of the trade societies, sometimes in pure ignorance of the real meaning... best example the Freemasons, who just took over the rituals of the Mason's Guild.