I'm no authority, but I have a hypothesis.
Measuring weight is more complex than measuring volume. Before the invention of the digital scale (recent history), or the spring scale (1770, by a Brit) things were weighed with a balance scale and a set of weights. Materials alone makes this more expensive than a simple cup that would hold a liquid.
Given the timing of the invention of the spring scale, and our subsequent armed revolution, and many years (decades?) of being a flat-out poor country I don't imagine that we had much money to worry about spring scales or balance scales. A cup is cheaper and simpler, and doesn't wear out or break. The durability also likely had a role to play in our journeys westward.
Likely by the time we actually could afford "fancier" methods of measurement the volume thing was just too ingrained in our heads. Plus, we tend to be stubborn (e.g. metric).