> A side effect of the geometry simplication is that there are some very small gaps between states. Based on your use case, you'll need to handle the case of the point not being within any state borders. In these rare cases, you could fall back to a different method, such as distance checking centroid points, adding an episilon to all state borders, or simply asking the user. (The user may also be in another country or in the ocean...)

This is a common topic and easily dealt with by working with topology-informed geometries; most simplification algorithms support topology handling between different features. For instance, TopoJSON can be used.