How many do you need though? If you take 0.1-1% of your testing capacity, those handfuls of samples a day give you some information. Not regional level or very reliable data, but it gives an idea of whether people (as a population) are having it and are having mild symptoms and are not reporting anything, or if it isn't widespread yet and the hospital cases are the only cases.

Aldo if you expect the true positves to be a small percentage (which should still be the case in most countries), false positives will dominate in a ranom sampling. Apparently false positives are high for unsymptomatic cases. Of course if you can estimate the false positive rate well enough, the random sampling might still give useful statistics.

As far as I know lab performed PCR tests are assumed to have 0% false positives and about 29% false negatives. Antibodies tests have about 10% false-positive rate but to the best of my knowledge countries in Europe widely use PCR tests.