You're simply wrong.

Even if the healthcare system eats the cost (as it would in my country), it's not supposed to. That's akin to entering a health insurance contract and then demanding the insurer pays for the theft of your iPhone.

Road traffic is insured by car insurers. The Uber driver doesn't have any, the car insurer therefore doesn't pay, so someone else pays. It doesn't matter if you rationalize it with "oh, that someone else is also some insurer". It is someone not involved at all. Ergo an externality.