Quarantine is not only done to observe the bird and determine that he has no symptoms of disease or that he doesn't develop any (birds get sick when stressed out and rehoming is very stressful to them) and to prevent possible contagion although those two are, most definitely, the main reasons. The other two reasons are that the bird needs time to become comfortable with its new home, schedules, human and other birds PLUS it needs to convert to your own diet and that takes time which quarantine provides.
As to testing for contagious diseases, nobody actually tests for every single thing there is out there. It's hugely expensive and nobody does it. And out of the common diseases, chlamydiosis and Pacheco, for example, require a number of samples collected at different times because birds with chronic forms don't always shed the pathogen so unless a series is done, a single test with a negative result doesn't really mean the bird is clean.
And you can't really trust what you are told. People lie all the time about what they do and don't do to their birds and it doesn't matter if the person you are getting the birds from is somebody you know and trust because this person might have the wrong information, too.
No vet will ever be able to tell you that your bird is 100% healthy, because nobody in the world will put a bird through ALL that testing. They can do the normal testing that covers a few common illnesses but that still leaves hundreds of other illnesses that the bird is not tested for. That is why we still have to quarantine. 
They are having their checkups now that I've found an avian vet.
