I’m trying to create a real-life “game” using some contour maps. Is there any way to do this without trying to single out the contour lines and guess at how to even start to make a height map?
From a pure contour map on paper? That is going to be really, really hard. If you have access to the GIS data, you would be in much better shape. Ideally, you would get it in DEM.
Photoshop… paint it in greyscale per height (white max height, black min)… make it obviously step… Then blur the crap out of it 
Out of ideas…