Why is the expertise they have built up in car-hailing directly applicable to other logistics problems, and what advantage will they have going in that creates a barrier to entry for newcomers?

I think there is one answer to this for Uber, and its the same strategy Amazon stumbled into: have your "main" business be breakeven, but provide the scale needed for a pure profit business. In Amazon's case, the main business is retail and the profit center is AWS.

In Uber's case, this would look like individual ride-taking customers providing the demand to keep a critical mass of drivers on the road. This business is breakeven at best, likely slightly money-losing (sorry folks, but an increases in ride costs will quickly stamp out demand to pre-Uber days).

Then, with this "infrastructure" of drivers on the road, they can leverage it into nearly pure profit areas like last-mile delivery.

There's a huge catch, though. Amazon just has to buy new machines every 3-5 years. Uber's driver turnover is much much higher, and is absolutely killing them. All the claims of profitability are deceptive, because they assume current demand with current set of drivers; those statements do not factor in driver recruitment costs next month, or tomorrow.

Unfortunately this appears on the surface to be a structural problem. Current pay with driver-owned vehicles is probably not a long-term or full-time possibility for most candidate drivers.

Edit: to save a step, the predictable response to my objection is always, "but... self-driving cars!"