Do not know about graves or hashimoto testing, but with celiac:
it's not that the antibody test is INaccurate so much, as it depends on how far-reaching the damage has penetrated.
The damage starts in the small intestine; so the most accurate test with celiac (and the earliest indicator) is to literally have your stool (poop) tested for atrophied villi.
An excellent lab is located in Dallas, TX.
They also send test kits through the mail: https://www.enterolab.com/
Only if the damage continues, and penetrates into the bloodstream, will you see the antibody test more accurate for celiac. I'd rather have my stool tested than have a surgeon puncture my small intestine and biopsy me, wouldn't you?