"Exercise helps post-COVID cognitive decline and insomnia" is absolutely bro science, and especially since it can make the whole situation much, much worse.
And a cursory look at the literature for PEM[1] will show you that the recommended path to recovery isn’t “avoid physical exertion at all costs”, but to do what you’re capable of and slowly ramp up the intensity, in which case small amounts of exercise could play a vital role.
[1] http://www.phsa.ca/health-info-site/Documents/post_covid-19_...