You do see how that's different, right? Me, contracting a sole individual to perform a task.
I do not have the means to completely vet the person I hire. To ask me to do so is a bit insane. Especially for a one-off trip. Not to mention, the driver in your hypothetical would be the company providing the service.
I mean, look, you have to put all these caveats on your hypothetical. I can't hire a taxi, because then the taxi company would be liable. I can't Uber, because that's a whole mess of who's liable. Etc.
Things are different from me personally hiring a single ride to the airport and me operating a transportation company to shuttle people to and from the airport.
If someone paid me to find them a ride to the airport and the person I hired to do the job wound up running over a kid, then I would be liable. Because then I've taken on the responsibility of vetting the driver.