I don't know, the world's 2 most ancient professions are still around... ;)
Actually, I'd imagine that truck drivers, especially long haul, would suffer before taxi drivers.
The taxi 'industry' would be fine, perhaps even invigorated by this. Fire all the nasty taxi drivers, have a computer do dispatch, etc...
Reduce costs enough and people will be less likely to buy a car rather than just renting one when they need it.
As you say - need a heavier vehicle, rent one, even over the phone. Heck, buy something and arrange for delivery. It'll drop the package wherever. Though delivery services at that short range are often cheap enough to be worth the human labor placing it in it's final location and hauling the old one away.