It depends on the state laws and the people involved. In many states, the bitten person can opt for either the quarantine or euthanasia/testing. So many people choose the quarantine but they can force the issue testing if they're bitten (in some states).
It depends a bit on where someone is bitten, too. The virus travels through the nervous system to the brain. If you are bitten, for example, on a hand or foot you have time to wait for that 10 day quarantine to be up because it will take longer than that to reach the brain. So even if they aren't showing symptoms at the time of the bite, if an animal is shedding virus at the time of the bite it will start showing progressively worse symptoms in plenty of time to still get treated, well before the virus has reached the brain. If you get bitten somewhere closer - say on the face or shoulder, you may not really have that luxury.
Also, pre-exposure vaccination for rabies varies a lot among veterinary professionals. All of our doctors are vaccinated, but a few of our technicians have chosen not to be, and I don't think any of our reception staff is. So if I get bitten I really don't worry about it too much, ever, for myself.
Seems weird that the clinic tested it themselves, though. I always thought an official state lab had to test them as well.