I have indeed taken a taxi before. I still have the photo in my album of cherished memories. I flip back to it each morning to relive the experience. Ahhh.
"Nearly impossible" is a simultaneously weak and strong claim. Let's try to break it down.
It may be useful to distinguish between shared rides and people using Lyft / Uber / X for a single group riding to a single destination. If there are stats on the distribution of number of parties in a single trip for the major providers, that would be a helpful stat. My anecdotal experience is that most rides I've taken and that I've seen others take are not shared, particularly for those on the higher end of the willingness-to-pay spectrum mentioned in the article. (If you're on your way to an important meeting, do you really want to throw in the variability of maybe picking up someone else along the route?) I don't think Lyft and Uber are perceived primarily as shared-ride services -- they originated as alternatives to single-party, single-destination rides. IOW, as direct alternatives to existing taxi service.
In terms of efficiency, please look at the breakdown of costs in the article -- I even excerpted some of those here. The cost of dispatch is actually quite low among the components of costs to run an urban hail-ride fleet. Look at Uber's R&D costs and, even amortized over the anticipated rides, and compare that to "a guy on a telephone." One fully-loaded Uber engineer's salary would pay for tens of full-time dispatchers in N. Amer. / European cities and hundreds of dispatchers where labor costs are lower. Similarly, the costs of fleet operations in terms of cars are going to be higher. Do you think "some driver with an app" can maintain and utilize a vehicle better than a taxi fleet, which can run the car 24/7 and have centralized maintenance facilities and bulk discounts on supplies and fuel? Efficiency considerations need to take into account both the costs and benefits. I think one of the points the article is trying to make is that Uber has been hiding these costs or relying on subsidies from private investors until recently.
One of the broader points the article attempts to make is that urban ride-hail fleets have attempted to optimize for things other than the pure efficiency of the rides. For example, they optimize for serving otherwise underserved areas (where some taxis won't go or won't take you because they're concerned about the unpaid backhaul). They optimize for some driver protections. I think it's useful to consider what is going into any definition of "efficiency." Is it utilization time of the vehicle? (This doesn't account for number of passengers served if the rides are longer because the rides are in suburbs rather than dense urban cores. It also doesn't account for taxis being utilized 24/7 while a privately-operated Uber maybe is operated 14 or so hours / day -- if the driver is willing to push the margins of safety and sanity.) The number of rides per hour? The time it takes per unit of distance? (On that last point, there's some evidence that the rise of uncapped ride-hailing services in urban cores has led to significant traffic increases s.t. it's actually slower to get around now in dense cities precisely because of Uber.) The article attempts to call attention to some of these other possible dimensions of optimization. Depending on which dimensions are optimized and the transportation network, it is entirely possible for "a guy on a telephone" to be more efficient. And questions of fairness -- for drivers, for pedestrians and mass transit riders, and for others -- will be affected by that choice of optimization.
Uber could be profitable at higher prices. But would we just be left with a higher-priced taxi service that significantly decreased fairness?