I also wouldn't worry. Not to disparage the hard workers who administer routine school health screenings, but basic school testing equipment is often over-used, out-of-date due to budget constraints, and sometimes not administered by a fully-trained professional. The point is to screen for potential issues, not to diagnose or determine causes of any abnormal test results. Of course, the school health staff does their best, and they're some of the hardest working people in our schools, but their job is to refer anything out of the norm, and it sounds like your son got the best possible school evaluation.
You did the exact right thing, which is to have your child examined by an audiologist. My daughter recently had a complete audiology exam to rule out ear problems (it ended up not being an ear problem but a type of migraine called vestibular migraines which cause vertigo, which was confirmed by the neurologist to whom the audiologist referred her). She sat in a booth and listened to different tones in different ranges and pushed a button when she heard the tone. The booth was completely sound-proof and the equipment was all the latest. Then the audiologist put special headphones on her and said a variety of words, which my daughter had to repeat. The audiologist did a couple of other tests, all non-invasive, and painless. She also had a thorough ear exam by an ENT specialist, and a couple of other high-tech tests.
There are so many reasons for failing a test: it could be the fault of the equipment, or an infection, or tonsil problems, fluid or wax build-ups, swelling after a virus, or so many other things.
After the audiology exam, the audiologist will prepare his or her report, and usually an ENT specialist examines it and makes recommendations, either for more physical exams, perhaps a CT scan, or whatever is appropriate. You probably won't get results right at the audiology exam, because all the test results must be compiled and reviewed, and recommendations or follow-up referrals prepared. In our case, we had an appointment about a ten days after the audiology exam when the ENT specialist went over all the tests with us and told us that migraines were causing her vertigo, not an ear problem.