You are correct. The increased numbers of people infected is not because of testing. Its because of counting 15+ people around someone that only shows a symptom or two, like the states are being told to do now.
My personal doctor told me they actually do the swab if they show a sign and only 45 people out of 14,000 in their experience are confirmed cases that's 0.3% infection rate.
However more testing would show an increase if more people were infected than the current known rate.
Its not a fallacy. Its basic logic. Let me explain again:
If you have 100 people, and 20 of them are infected but you've only tested 20 of them and show 1 of them is infected. Then later you test the other 80 and find that 19 of those are infected, you get a higher infection rate.