From my own experience and what I have heard from family and friends, it is very common to be carrying the virus, but the first tests are negative. Then, a few days later, the positive test kicks in.
It works in reverse at the end of the illness. You can be feeling perfectly fine, definitely over covid, but the tests keep saying positive, sometimes for weeks afterward. That happened to me. I was mildly ill for 2-3 days, mostly just sleepy with a bit of a cough. Then I felt fine. I tested every day, and it took two weeks before the tests finally came out negative!
My whole family (all vaccinated aside from the babies) came down with it at Thanksgiving. We all had a similar experience. So the test does show something about whether you have it or not, but the timing seems to be off. Better than nothing, I suppose. Probably the tests will improve in their accuracy over time.
But whatever the tests do or not show, everyone who medically can be vaccinated should be. No ifs, ands, or buts. Againstthegrain summed it up very well.