There are pre-Columbian Exchange starches still used as thickeners in Chinese cooking. They're less common today than potato starch (which is very common in China; Westernised recipes will often substitute corn starch for potato starch (and also for other less supermarkety starches)) and corn starch, but still common enough to be sold in stores in Chinatowns across the world. These include kudzu/kuzu/arrowroot, water chestnut, lotus root, and mung bean. And, of course, wheat and rice starch.

Some things should be noted:

1. Starch-thickened sauces are much less ubiquitous in Chinese cooking in China than in Westernised Chinese restaurant cooking in the West. Westernised Chinese restaurant cooking in the West is largely a post-Columbian phenomenon, and has no significant pre-Columbian history, if any. Also, starch-thickened is less common in home cooking than in restaurant cooking.

2. There are non-starch thickeners. For example, mallow leaves will thicken soups (I must confess that I've never cooked with Chinese mallow, Malva verticillata, and don't know how it compares with melokhia (AKA variant spellings and Jew's mallow), Corchorus olitorius, the usual African and Middle Eastern mallow as a thickener), and sesame paste and almond paste were used historically. Sesame paste is still used today in Chinese cooking (and also does a splendid and tasty job as a thickener in West African cooking). Bean pastes (both fermented soy bean pastes - which often contain wheat - and Sichuan-style broad bean pastes) significantly thicken dishes.

3. Sauces can be thickened by reduction. Just cook them for a little bit longer.

4. Some sauces (bean pastes, oyster sauce, etc.) often have wheat as an ingredient, and the what starch will act as a thickener when these sauces are used.

Also, there is another common Columbian Exchange starch used in China: tapioca (cassava) starch.