Antibodies build up over time. So if you were to test for them right after you are infected, you might not have any that show up on a test. A week later, the level might be high enough to show up, but not consistently. Two weeks later and the antibody tests get up to 98-100% accuracy.

Also, there are two different types of antibodies (IgM and IgG). The IgM are short term right after the infection and then drop quickly and go away. The IgG are long term that provide lasting protection and are built up based on the template of the short term antibodies.

As for the tests themselves, the FDA has removed a bunch of bad quality ones, has given emergency use authorization to a few and is still reviewing others.