The reason that relatively untrained cryptanalysts are able to use frequency "on their own" is because the civilization has advanced enough for them to think in a way that allows for them to  think in complex ways. Though they are relatively untrained in the discipline of cryptography, they are well versed in the realm of linguistics, both reading and writing, and as well as at least a basic understanding of math and statistics.  This is very different from the civilizations that Singh refers to when discussing civilizations that weren't advanced enough to to preform crypt analysis, in those civilizations writing, reading, math and statistics were reserved to such a select few that new and original ideas were hard introduce. Because there were no new ideas that came into the intellectual aspect of the civilizations and the intellectual community is so small, it is hard for people to notice the patterns that is the basis of frequency analysis. In today's society, even those that are classified as "amateur" cryptanalysts still have a solid base in linguistics and at least a basis in math and statistics which allows them to be able to come up with frequency analysis "on their own."   The intellectual community is larger and because of that , there are more examples of cryptography and as Singh shows in his book, the more material there is, the easier it is to see the pattern of the encryption that hides the plain text of a message. The intellectual basis combines with the fact that the intellectual community is larger in order to create the opportunity for frequency analysis to appear naturally in today's society, as opposed to the civilization, such as Europe, who Singh classifies as not having "sufficiently sophisticated level of scholarship ".