Does a viral load at the levels the article calls a false positive still trigger the production of antibodies?

Of course not. This phenomenon us known since May, my estimation is about 20% of such false positive PCR tests. It's getting cold, people are getting symptoms, and the rate of asymptomatic cases is 90%. So a lot of people are getting tested now, but had an infection months ago. The PCR tests still picks up the pieces now.

Every PCR positive cannot be called new infection, because both might be false. Not new, and not infected.

That's why everybody is putting the much better and cheaper antigen tests forward. Only when that is positive, a PCR test needs to be done. Antibody tests are highly unreliable, but antigen tests have no false positives. China tested a whole 5 mill city in a few days with antigen tests, Slovakia tested their entire population in two weekends.

So people have the virus in their bloodstream, but it doesn't do anything and the immune system doesn't react either? I don't understand.

The PCR test is overly sensitive. The virus has a ~32k DNS, the test searches for len=600 matches. If the immune system has killed the virus the remaining RNA strains in the body usually are much longer than 600 even if the virus is long killed, and torn apart. This not infective. It was found out that too many people tested positive again at the 2nd test. This was not a reinfection. The reinfection period starts after 6-12 months, as with any other common cold.

The best test for testing infectiveness is the antigen test, and then you need a 2nd PCR test for confirmation that it is SARS-COV-2 and not some other common cold variant.