Wouldn't meningitis be detected by symptoms in someone who was being tested for it?

Suspect, not confirm. You might suspect it, but since a headache or stiff neck can be caused by a myriad of things you would never confirm or treat it without proper testing. The lumbar puncture used to confirm it also provides access to information on the germ.This allows one to decide whether it is viral or bacterial/fungal and refines treatment choice.
Most often. It certainly diagnosed by the clinical exam but the real answer lies in the spinal tap because not all exams particularly in infants can detect meningitis. The spinal tap will confirm it and direct treatment.