It seems like the Yale report is actually a rebuttal to what you are saying. The hospital my wife works at uses the Abbott test in their ER and as you can see it is actually a very low sensitivity test with a Ct ~= 20. I'm not saying that false positives exist, in fact quite the contrary but also that there are methods to accurately assess whether a patient is infact infected with COVID. I think the truth lies somewhere between "the tests accurately asses covid infections" and "this test is garbage and always positive".

From the yale report: • Patients with pneumonia may have little virus in upper airway and using a sensitive assay is essential." - pg 9 • If less sensitive test was used, diagnosis would have been missed and patient discharged to SNF as COVID negative" - pg 10

Highly sensitive tests are essential for acutely ill hospitalized patients as virus titers in the upper airway may be low (Ct >30 or Ct >35). However, recovering patients, now non-infectious, may also have a very low positive PCR result. • For diagnostic testing in the community, delays in obtaining testing, as well as sample type and quality, can lead to higher Ct values at diagnosis. Not reporting positive results with Ct >30 would be a disservice to these patients. • Reporting Ct values alone can be misleading, especially since Ct values can vary significantly between various tests and labs. However, a result comment for low positive results may be helpful. Ct values >40 may be of questionable value. • It is essential to confirm actual test sensitivity, determine the goals of testing and understand the tradeoffs in various groups: e.g. asymptomatic screening, symptomatic patients, pre procedure, L&D, high risk nursing home residents. • Tests with rapid but somewhat less sensitive results may be acceptable in some outpatient settings, especially when frequent repeat testing is performed. - pg 14