Why do amorphous substances have different values of physical properties in different directions?
Yes, you’re exactly right that we are talking about an average. If we could somehow look at a small enough portion of an amorphous solid, we would see differences between the properties in different directions. Indeed, a small enough portion of, say, amorphous ice is a single water molecule, which is certainly not isotropic: we can talk about the direction perpendicular to the molecular plane or whatever.

But when we talk about real, physical materials, as usual in chemistry, we are talking about Avogadro’s number (plus or minus a few orders of magnitude) of molecules. So taking the spatial average of an amorphous structure makes perfect sense, and the law of large numbers means that the tiny differences between properties in different directions will be utterly undetectable.

(As an aside, it’s worth noting that the perfect crystalline structure we imagine is also an average over every unit cell in the material – it’s just that because each unit cell is oriented in more or less the same way, different directions are not “mixed” as part of this average, so crystals can have anisotropic properties.)