https://www.healthline.com/health-news/false-negatives-covid...
The antibody tests are a different animal. They test to see whether you might have been infected in the past, by looking for antibodies in your blood. So these are blood tests, and, I expect, have less chance for user error.
But also, I think lots of covid antibody tests are not high accuracy even "in theory". It seems the FDA relaxed oversight on companies producing antibody tests: ". . . after being criticized for the fumbled rollout of diagnostic tests during the start of a global pandemic, the FDA swung hard in the other direction, waiving its usual requirements and letting firms rush self-validated [antibody] tests into the market."[1]
So there are antibody tests from over 200 companies out there. Some of these antibody tests are quite accurate, high sensitivity and specificity. Others, not so much.
[1] https://khn.org/news/hype-collides-with-science-as-fda-tries...
This is not the nasal swab test that sometimes misses positive people, as you mentioned.
To avoid confusion over this issue: There is an Abbott nasal swab test, that gives results 15 minutes after being swabbed, and without "special equipment". That specific Abbott test (the instant nasal swab) has experienced dubious results in practice, even compared to more standard nasal swab tests.
What is important is that antibodies are a sufficient but not necessary indication of past infection. There seem to be plenty of people who don't develop them when exposed, possibly due to effective T-cell immunity.