Siddham did not cross into East Asia in one step. It moved with Buddhist texts along the overland routes through Central Asia and by sea around the south, through a chain of monasteries, oasis towns and translation bureaus in which Sanskrit, several Central Asian languages and Chinese were all in use at once. The manuscript caches recovered from that corridor contain material in a startling number of scripts and languages side by side, which is the ordinary condition of the places rather than an accident of preservation.
The translation bureaus are where the script became necessary. Rendering a sutra into Chinese meant deciding, phrase by phrase, what could be translated and what had to be transcribed, and the answer for mantras and dharani was that they had to be transcribed. A bureau doing that work needed people who could read the Sanskrit original, and the Chinese Buddhist canon preserves the by-products: compendia of the letters, tables of their sounds, and manuals explaining how the script is put together.
Japan then received the whole apparatus at once, and received it late, which is part of why it survives. The transmission Kukai and his contemporaries brought back in the ninth century was a package of texts, images, implements, lineage and script, assembled in Tang China from centuries of prior work, and it was carried to a place where it was kept together rather than dispersed.