By the way, I really don't get people's complaints about "predatory pricing".

Let's say Uber is charging a really low price for their service, much less than what taxis think they can charge. Sounds fine to me. Low prices for customers!

Let's say it's so low that Uber is losing money on every ride. Well, obviously this is an unsustainable strategy. (I've heard that this is happening in some places, and my interpretation is that it's essentially marketing, to get people in the habit of using the service and working for it.) I don't know, maybe there are economies of scale that will kick in and it'll become slightly profitable for Uber at the same price? Long-term low prices for customers: sounds good to me. But let's suppose not: eventually Uber has to raise prices so they're making at least a bit of profit.

Once they do that, the taxis can presumably charge the same price and compete. And we get long-term low-ish prices (lower than the original taxi price) to customers. Right? Unless taxis' costs are higher, such that the price in question is slightly profitable for Uber, but would give a loss to the taxis. Well, then, Uber is more efficient, it'll remain around while the taxis die, and customers get long-term low-ish prices. Sounds fine to me.

If Uber subsequently jacks their prices way up, to what taxis' prices used to be or higher, then taxis (and Lyft and similar companies) can come back into the market, charge the old price or lower, and make a profit. Unless they're all colluding to keep prices high, or getting regulators to force the entire industry to so collude, competition will eventually force the price down, and most likely customers will get long-term lower prices.

The only way I can imagine a "predatory pricing" scheme that leads to high prices to the customer in the long term is if "network effects" and "economies of scale" mean that the market leader has a large advantage in supply costs; the industry might then be described as having a "natural monopoly". Then Uber might spend a lot early on to capture the market and then sit on their advantage making a large profit while charging medium prices (which are roughly break-even for competitors). Even supposing that's the case... what's the alternative?

Right now, either taxis are one big monopoly and already gaining from the network effects and economies of scale, in which case it's just one monopoly replacing another... or they're not, taxis are fragmented and only colluding insofar as they're restricting supply but not cooperating for efficiency gains. Even if it's the first case, the barrier to entry of "you don't have network effects and economies of scale" is certainly lower than "the government will shut you down", and I'd expect a correspondingly lower resulting price. In the second case, it's better, because even if Uber owns the entire market, it will be in their economic interest to charge a lower price (and produce more of the service) than was in the best interests of the inefficient taxi cartel. This is a general fact of economics:

For a firm with a monopoly, there is an optimal price they should choose to maximize their profit. It will be a higher price and lower quantity supplied than the break-even, maximum-supply result of "perfect competition"; at the optimal point, there are additional units that the firm could have produced and sold profitably at a slightly lower price, to additional customers who would have bought them--but without good price discrimination, they would also have to lower the price they charged for all the earlier units they sold, and the optimal point is where these effects exactly cancel out. Now, if the firm's cost of production is reduced--specifically, if the marginal cost at what used to be the "optimal point" is cut down, then that calculation about the marginal profit of the next N units vs the drawback of having to sell everything at the lower price gets pushed in the direction of "marginal profit", and thus the optimal point moves to a lower price and a higher number produced.