Sure, nobobody is saying that anxiety disorders aren't a mental illness. But it's also likely the reason why the rate of mental illness was found to be 1 in 5. From your own link, 1 in 5 adults already have anxiety:
>Over 40 million adults in the U.S. (19.1%) have an anxiety disorder.
So it follows that undiagnosed anxiety disorder would be found in abundance in people recently diagnosed with covid.
It looks like they accounted for that or at least did as much as they could to account for that.
>In patients with no previous psychiatric history, a diagnosis of COVID-19 was associated with increased incidence of a first psychiatric diagnosis in the following 14 to 90 days compared with six other health events.
So it looks like they compared it to six other health events and it (anxiety, insomnia, dementia) was still higher with COVID, but they add "we cannot exclude possible residual confounding by socioeconomic factors."
>Over 40 million adults in the U.S. (19.1%) have an anxiety disorder.
So it follows that undiagnosed anxiety disorder would be found in abundance in people recently diagnosed with covid.