If it's a regular screening test rather than a response to the patient's complaint, then false negatives wouldn't be as much of a problem because at least it's better than nothing.
> it's better than nothing.
Not necessarily, that's the issue with screening asymptomatic people. You have to balance the consequences and rates of false positives with the benefits of true positives. If ultrasound screening mostly catches indolent diseases, or those where catching them so early doesn't affect outcomes too much; and you have a lot of people going through unnecessary/risky tests, you might end up doing more damage with those screenings. It's not so easy.
Tl;dr: the first diagnosis tends to stick with the patient.
Newborn babies undergo a whole lot of simple and inaccurate screenings. I guess those are cases where the harm due to inappropriate treatments is lower than that due to leaving diseases undetected.
Either way, I'd rather the technology exist and then people work out how best to use it, rather than the decision being forced on us by it not existing.