For tests in the real world, there are always false positives and false negatives. No test is 100% perfect. The question is always how high those rates are, and thus how well you can rely on the results.

The antibody test has a failure rate of 10%, the PCR test a failure rate of <0.1%.

In civilized countries the PCR test is free for critical cases (like travelling abroad or in contact with a positive), or 60 EUR if not.

Sure, but if you had symptoms AND test positive, it is quite likely (above 95%) the result is correct.