>demand is hard to forecast for an individual, almost completely inelastic and often extremely time sensitive
These don't preclude a free market working well. For instance, they're all apt descriptors of me when I find myself needing an Uber home. I'm not completely inelastic, since I could take a series of long bus rides to get home, or walk, but accepting for the sake of argument those aren't life-threatening, the cab companies more or less have me over a barrel. Or do they? We all know what keeps them from charging me more they do, even given the time sensitivity, lack of prediction, or inelasticity. It's competition.
The USA system doesn't work well, and I'm not necessarily saying that free market is the right solution, just pushing back on the notion that just because a good meets those criterion we are forced to throw our hands up and say the free market could never provide that.