Do you mean "Do their own research" like make up conspiracy theories out of thin air and parrot nonsense deep state rhetoric? Completely divorcing yourself from reality? You want more of that?
Skepticism of any claims is a pretty good heuristic, but should be equally applied to counter claims.
Everyone would agree that a person wandering around the street firing live rounds into targets is acting in a dangerous manner.
No-one would agree that a person drinking a cup of tea in their own house is acting in a dangerous manner.
But coronavirus isn't one of those things. It's genuinely a borderline issue, and not because of the science, but because it's almost exactly on the line in which some people consider it to be a disaster and others consider it to be a bit of a nothing burger.
As a result, you end up in a situation as described by rgrieselhuber below.
Some people think that we need to close (e.g.) schools to stop coronavirus. Other people think that we need to stop caring so much about coronavirus, so that we can go to (e.g.) school.
The question of whether there's misinformation is almost tangential to the point because there are huge numbers of people who do not doubt official statistics or modelling but also believe that data or those models are not severe enough to require us to 'stop the world' so to speak.
I guess my thoughts are that, if we can't even convince each other (and from what I can tell neither of us have really fallen for misinformation), what hope is there for the actual 'vaccines cause 5g' nutjobs?
What does this have to do with the price of fish?
Moreover, it's not just about the media's coverage of COVID or the moon landing, but about their overall credibility during the prior decade. For example, the media has reliably and brazenly lied and mislead on racial justice protests from ~2014 through 2020, which almost certainly affects its credibility on other issues (e.g., covid). Of course, the same applies to the academy and other institutions.