This is largely incorrect. Outside of Han China and a few other areas in its historical cultural area of influence, alphabetic scripts won over ideographs, and won big time. The Egyptians developed paper-like papyrus thousands of years before the first uses of "paper" paper, and they largely abandoned their ideographic hieroglypics in favor of the Heiratic and Demotic scripts, which are syllabaries, for all but formal usages.

Now if the Mongols or the Khitans or the Uygher steppe empires had been able to impose their scripts on the Chinese after conquering them, instead the assimilation going the other way, Unicode might be a considerably smaller clusterfuck today.