The lab confirmed that they had processed all of the priority tests with time to spare, because they were accused of not having capacity for priority tests, and under that accusation/assumption, the second fingers were pointed at WADA for not certifying another lab to process Rusada's samples to be sure they were processed in time. And the lab had to clear its name, when the timing of Valieva's results showed that the IOC and/or WADA had clearly not confirmed that they had negative results for all of the Olympic athletes going into their events, and they let at least one athlete compete without having results present. So they had egg on their faces.
It is possible that they let other athletes compete without negative results in the first few days, while scrambling to confirm that there weren't other missing results, or for the rest of the Olympics, since it wasn't made public, but it wasn't because of this particular lab's priority processing record. So, yes, not marking it priority was the reason why this lab didn't process Valieva's sample until three days into the Olympics.
Unless, of course, you can show that the lab was lying.