My mom is a pathologist for a major metropolitan area hospital, and working at this moment in time as I type this.
Microbiology was my field of study before I decided to pursue music.
Regarding testing, at this moment it's a capacity issue.
The assays required and analysers involved in the laboratory testing of specimens are not perfect, but they are reliable enough, given the right parameters...a person who is symptomatic, and tested, within the clinically relevant date range, will produce a positive result in the laboratory, almost every time, less so, if it's early in the exposure and incubation period, or vice versa.
Furthermore, for inpatient care, several different protocols for laboratory testing include redundant and complimentary use of a direct draw (vial of blood) test which requires the sample be spun, and then prepared for an analyser to determine the presence of antibodies, as opposed to antigens.
A positive antibody result indicates the patient / specimen was infected with covid, and has developed a stored immune response, so to speak. There are additional markers which can indicate infection.
There are only so many techs and labs and machines, and the laboratory infastructure can only process so many. There are numerous reasons why our testing capacity continues to lag. There are systematic and institutional failures of course, but the assays and analysers involved are fairly reliable; Humans, and our healthcare systems, are not as reliable, unfortunately.