RE RANT: It all depends on what you are trying to do. One of the most common applications of triangulation is to connect sparse/irregular samples of a continuous function (well water depth, for example) into a surface, which is more convenient with the default behavior. In that case, you can think of the color at the vertices as the sampled value of the function. trisurf with 'Edgecolor', 'none', 'facecolor', 'interp' gives about as credible a version of the raw data as is available. When the points aren't too close together, it can be useful to overlay a plot of dots at the sample points. You could, of course, interpolate the data to a fixed grid (interp2) or form a gridded surface with gridfit to beat down sampling errors, but neither will preserve the full fidelity of the raw samples in a way that scales as you zoom in, which trisurf will. All that said, any default can be a problem when you are trying to do something else. Ergo, options... which might be a bit lacking for your desires.
Curious: What is it you are doing that the Cvalue is a characteristic of the surface, rather than of the verticies (as above)? Depending on the answer, you might be able to use the vornoi diagram dual, which has a vertex to the circumcneter of each face, and triangulates those. In essence, you could interpolate from face to face, rather than from vertex to vertex. This does lose information about the face edges, so it may or may not be suitable for your needs.