The Chinese Buddhist canon is not only a collection of sutras and commentaries. It also carries the working apparatus that the translation effort produced, and part of that apparatus is a set of compendia about the Siddham script itself: tables of the letters, accounts of their sound values, explanations of how the vowel marks attach and how consonants combine, and lists of the conjunct shapes. They were compiled in China during the centuries when Sanskrit texts were arriving in quantity and monks needed to be able to handle the originals.
Those compendia are the channel through which the script reached Japan as a subject rather than as a curiosity. A Japanese monk studying shittan was in large part studying this Chinese literature, and the Japanese tradition of Siddham scholarship grew directly out of it, adding its own manuals, dictionaries and commentaries over the centuries that followed.
Their survival is a piece of good fortune with a plain cause. The Chinese canon was copied, catalogued, carved into blocks and reprinted repeatedly by institutions with the resources to do it, so material that was never popular survived because it was inside something that was. A great deal of what can now be said about how Siddham was read in East Asia rests on documents that were preserved as part of a set rather than for themselves.