It's not really that simple, barycentric coordinate access is relatively recent. It's asking the rasterizer for information and transforming that information into barycentric coordinates, and the correspondence of barycentric coordinates to vertices is unstable without further hardware support or further shader trickery. In the case of AMD gpus, it's only RDNA2 and later that have hardware support for stable barycentrics.
And you're right that this has been thought of. There are other approaches for bilinearly interpolating quads that have been historically used, but they require passing extra data through the vertex shader and thus often splitting vertices.
And you're right that this has been thought of. There are other approaches for bilinearly interpolating quads that have been historically used, but they require passing extra data through the vertex shader and thus often splitting vertices.