Actually, yes it should, but it depends on the numbers and this is a well studied area of statistics.

One of the many issues is whether or not someone who flags positive is actually positive and whether someone who flags negative is actually negative. This depends not only the sensitivity (if someone is sick, will they flag positive) and specificity (if someone is healthy, will they flag negative), but on how prevalent a disease is within a population. That last quantity is constantly changing, which also means that this calculation is constantly changing, which can and should affect policy.

Now, the issue is that if you start telling a bunch of healthy people that they're actually sick and want them to quarantine or that a bunch of sick people are actually healthy and that they're free to go there are consequences. One, people will lose faith in the system and stop listening at all. Two, healthy people who work in an essential area can now not actually do their job. Three, sick people in an essential area are now going back to work and getting other people sick.

That doesn't mean that nothing should be done. However, it may absolutely mean that this particular method should not be done because it will cause more harm than doing the proverbially nothing.

As for this particular article, it's not great because it doesn't actually work these numbers. It would be a good article if it actually calculated whether or not the app is worthwhile given a certain prevalence, sensitivity, and specificity. Personally, I don't know rate of prevalence of COVID-19 nor the sensitivity or specificity of the existing tests. However, I still strongly reject the premise that any test or contact tracing app is better than none at all.

In order to fully beat a dead horse, here's the wiki that runs through how to calculate these numbers:

https://en.wikipedia.org/wiki/Sensitivity_and_specificity

I don't like their presentation, but it is what it is. Here is some Octave code that runs through their example with some more readable names:

% Total population
pop = 2030;

% If someone is sick, will they test positive
sensitivity = 0.67;

% If someone is healthy, will they test negative
specificity = 0.91;

% Prevalence of the disease
prevalence = 0.0148;

% Works some numbers
sick_pop = pop * prevalence;
healthy_pop = pop * (1-prevalence);
sick_pop_tested_positive = round(sick_pop * sensitivity);
sick_pop_tested_negative = round(sick_pop * (1-sensitivity));
healthy_pop_tested_negative = round(healthy_pop * specificity);
healthy_pop_tested_positive = round(healthy_pop * (1-specificity));

% Derive some useful quantities
fprintf( ...
['Of those who tested as sick, percent of those actually ' ...
'sick (positive predictive value): %2.1f%%\n'], ...
100*(sick_pop_tested_positive / ...
(sick_pop_tested_positive + healthy_pop_tested_positive)));
fprintf( ...
['Of those who tested as healthy, percent of those actually ' ...
'healthy (negative predictive value): %2.1f%%\n'], ...
100*(healthy_pop_tested_negative / ...
(healthy_pop_tested_negative + sick_pop_tested_negative)));

> Of those who tested as sick, percent of those actually sick (positive predictive value): 10.0%
> Of those who tested as healthy, percent of those actually healthy (negative predictive value): 99.5%