Why don't hyperbolic navigation transmitters on the surface (e.g. Loran-C) allow altitude measurement?
You are correct, the time difference generates hyperboloids in 3D space and as such can be used to find the 3D position. However, if the transmitters are far apart and/or in the same plane, the geometry of the intersecting hyperboloids is such that the accuracy in the vertical axis is very poor.

When the receiver is in the same horizontal plane as the transmitters, all intersections almost form a vertical line. The slightest inaccuracy in time difference, would cause only a slight error in the horizontal position, but a significant error in the vertical position. 

The concept is called Geometric Dilution of Precision (GDOP). A good start for more detail can be found on Wikipedia