Delta can't add more seats within minutes by increasing prices. Uber can. Drivers can and do decide to put in an extra shift very late at night New Year's Eve because of the rate premium. That's where the extra supply comes from. It's real. If not for this large effect I would agree with your analogy.
True, but there are very few last-minute airline award tickets sought (or paid tickets for that matter).
So the flexibility of the market for air travel may not be as instantaneous as it is for uber, but neither does it need to be. It's certainly flexible enough to respond to the market in order to offer 'market based' prices within the time-frame of a typical purchase rather than flat rates. Just like Uber.