You have too much confidence in that statement.
If false positive, false negative, and prevalence rates are all 5%, and let's say you subject 350,000,000,000 people to the dog test (ignoring the time it takes to do that and logistical difficulties with having enough dogs and handlers etc).
9.5% of society tests positive resulting 33,250,000 total positive results. Half of this number are true positives and half are false positives. That means, quarantining 16,625,000 uninfected people along with 16,625,000 infected people which results in the loss to society of their output and might also now get them infected depending on how they are quarantined.
Out of the 17,500,000 people who are infected, there will be 875,000 false negatives who will not be quarantined who will continue to spread the infection. In fact, 5% false negative rate sounds kind of optimistic. If false negative rate is 25%, 4,375,000 infected people will not be detected using this method.
You can play with the numbers using this calculator[1].
Second, let's say you put a dog at every company and every day everyone needs to be sniffed on the way in. If the false positive rate is 5%, the probability that you will not be quarantined due to a false positive during the year is 0.95*250 approx = 0.0003%.
[1]: https://www.covid2020.icu/false-positive-false-negative-simu...