Is tire particulate too big to show up in the AQI measurement? If so, does it fall out of the air so fast that it isn't really a problem?

The Aqi is an index which is guided only in part by Particulate Matter measurements. The index part was definitely designed before microplastics research without knowledge or regard to is effects. Pm100 doesnt register them, PM2.5 may register them, but grossly with any other particulates of that size - so you dont know if the captured particulates are microplastics or something else or something in between