The issue for the industry is not passenger confidence in their ability to travel without undue risk. It is the confidence of national governments in the effectiveness of anti-virus mitigations, in other words, confidence that aviation is not contributing to the spread of the virus. Once that is achieved, people will start to travel in the knowledge that they aren't going to be suddenly quarantined or similar. Pending a vaccine, preflight and/or arrival testing will be a key part of the process, at least according to the professors of epidemiology I heard speak recently; they also all commented that the speed and accuracy of tests is improving rapidly and that there are now products entering the market with accuracy in the 97%+ range.
Leaving antibody testing out of the equation (they only tell you if you have had the virus, not whether you have an active infection), the risk of a false positive result (for genetic material or antigen) is less of a problem than a false negative. The false negatives lead to infectious people entering the system, which then means you have to rely on the other mitigations to prevent transmission - those people who refuse to wear a mask, wash their hands, maintain social distancing when possible, etc. And for very small numbers who have a false positive there is always the prospect of a re-test; some can be done in a matter of minutes. And although any false results are unhelpful, the many correct outcomes suggest it is likely to be our best way forward.
I have a friend flying for a ME airline who tested positive on return to his home base and was ordered into quarantine, which surprised him as he felt fine and his departure test had been clear. A second test a couple of days later was also positive but he was still asymptomatic. A subsequent antibody test proved he had indeed been infected and recovered from Covid-19 even though he had no symptoms at any stage. The track and trace system worked out he had probably picked it up during his layover in the USA. Had he not been tested, the operator would have no idea about the risk at the US end, and the authorities would be dealing with all the people he infected after his return. That is the sort of process that boosts government confidence.