So you're saying the study is so shoddy that they didn't have a control group of people who went through the same lockdowns but didn't get COVID? Otherwise, your comment doesn't make sense.
You can't give a control group a "placebo illness" such that they don't know whether it's Covid-19 or not (and make it double-blind so that the people administering this illness also don't know).
The people who overcame Covid-19 had an experience that members of a control group who did not overcome Covid-19 do not have.
That's an obvious source of bias, which is inherent to the study.
What would be credible would be data about people who had Covid-19 but did not know it (due to mild symptoms and no hospitalization), showing that those people have mental problems compared to a control group of those who didn't have Covid-19.
However, if there is a real physiological effect there, it could be tied to the severity of the illness: i.e. how much damage it caused to the body. In that case, those secondary effects could be expected to be next to nonexistent in the mild group.
Your suggestion of another control group with Covid-19-but-no-symptoms would help answer that question too, though ideally we'd want those who had Covid19 but didn't know it at all.
But anyway, this paper is only establishing correlation: getting to causation in medicine is the harder task and takes years or decades and even then it's usually inconclusive!
What you want to control for is the lockdown effects or general disease stress, not covid-19 itself. That's what you're trying to measure.
Those aware they had Covid19 will fear all the known-unknowns around Covid19. Those who don't think they had it wouldn't have stressed about it at all.
But my comment was kinda tongue-in-cheek: How do you even find people not affected by this doom-n-gloom global news story?
No, they were studying the effects of COVID, not the effects of lockdowns. Your suggestion is controlling for the wrong variable.