Why, of course it can be done. Start with the net, assign the lengths so that all triangles will be equal and scalene:
Then fold it and glue everything together.
True, the figure looks god-awfully ugly. This is a consequence of the fact that it can't be made face-transitive, unlike your three examples. (To make 20 faces symmetrically equivalent, we'd have to keep at least one 5-fold symmetry axis, which would keep passing through a vertex, which would make its five edges equal, which in turn doesn't sit well with the triangles being scalene.)
If you agree on isosceles faces, then you may keep a 5-fold axis and arrive at Aretino's example.
I suspect there are multiple ways to connect the scalene triangles (this is the case with octahedron, after all), but this post is getting too long, so let's call it a day.