Are there any pre-modern/non-romanization examples of attempts to alphabetize Chinese?
Yes, there was such an attempt during the Yuan dynasty. The 'Phags-pa script was created for writing languages that were under control of Yuan, including Chinese:


  The 'Phags-pa script (Mongolian: дөрвөлжин үсэг "Square script") is an alphabet designed by the Tibetan monk and State Preceptor (later Imperial Preceptor) Drogön Chögyal Phagpa for Kublai Khan, the founder of the Yuan dynasty, as a unified script for the written languages within the Yuan. The actual use of this script was limited to about a hundred years during the Mongol Yuan dynasty, and it fell out of use with the advent of the Ming dynasty. The documentation of its use provides clues about the changes in the varieties of Chinese, the Tibetic languages, Mongolian and other neighboring languages during the Yuan era.