Uhm, I am not an expert but when I have retopologized something as exercise I have done it more manually than this. I have drawn the vertices with the magnet option, so that it projects my vertices right on the shape. For bigger area with no creases/details, I have used bigger regular surfaces like planes. And in the end I have merged all manually.
In the image, it isn’t trying, literally, to follow the bad topology, but it just follows the nearest points on the faces. When there are little details or creases it’s normal that it is gives this result.
So my suggestion is to retopologize it piece by piece, also because if you just put a sphere or a cube around it, it will be again a bad topology even if different because a good topology follows the flow of the mesh. Check some examples on Youtube to see what I mean. Just having parallel quads is not the right topology for a human (or not) face!
If you do it manually, and correctly, you can use subdivision surface modifier with success, after, and so avoiding to create too dense mesh with your own hands! Also, if you create a more similar shape around the model, using the shrinkwarp will be better. If you are interested on doing it, you’ve to create vertices enabling the magnet on the main toolbar and to select in its nearest menu the option “face” (so, where to snap the new vertices on the mesh). But before all see some examples on the web about the correct topology for a human.
If someone arrives with better answers, forget the mine.