Random thought experiment, but does this effectively mean that even if you just wanted to render a flat shape, there is no getting around making many polygons so that textures have high resolution? ie. if I had a singular square (two triangle) mesh I could have the shape of the US flag, but to render the stars well I need to create a physical mesh that outlines the stars? Is there potentially a nicer way to map textures within each triangle instead of only linear interp.?

The values of the colors are not interpolated based on the color values of the corners. For each point in the triangle, the texture coordinates are interpolated, and then the interpolated texture coordinates are used to sample colors from a texture map.

In this slide, the colors are just to visualize how the texture coordinates are interpolated; it is not meant to indicate that the color values themselves are interpolated.

Ah makes sense