Technically you are correct, however you are ignoring how these tests actually work in real life. Generally people with symptoms are the first people to get tested... the people showing up in the hospital with symptoms are getting tested before the people who have no indication they are sick.
1. If you test the 10,000 sickest people in a city you will find a certain number of cases. To get the covid-19 rate you divide this number by the total population of the city.
If you then test another 10,000... it is a mathematical impossibility that the covid-19 rate would go down (and the covid-19 rate will almost certainly go up as new cases are detected).
2. If you test the 10,000 sickest people in a city, the "positivity rate" is the number divided by 10,000
. The population of the city has nothing to do with it. If you then test the 20,000 sickest people in the city (you are now test people that are slightly less symptomatic), the positivity rate will very likely go down.