I try to distinguish between when I've researched something (as in gone to academic papers, source data, areas I've worked in with others to learn how to interpret the data) compared to read/watched something that had relevant information but not an area I know or something I've experienced which yes, sometimes I do trust more than something I've read in the news.
So with COVID, I would say I've read about the disease and vaccine. With the effects of COVID on schools in my area, I have a mix of experience and having done research by working with local Public Health and digging into the data sets. With the changing ideas around Long COVID, it's been a mix with more reading and a bit of personal research as I've interest and experience looking at data in that area from a 'why doctors categorize things as they do and why that matters' rather than a 'why these things can do this damage in that way' where I'm pretty limited.
Not COVID-related, I've had situations with medical professionals where I've brought in or emailed in sources to discuss with them. I spent 5 years misdiagnosed until I emailed in the NICE Guidelines on that diagnosis and asked to discuss with the GP at my next appointment about the treatments listed in it since the NICE Guidelines says I was being put at high risk by not being treated which caused that GP to look over my file to find out even with my many symptoms, my repeated test results were not positive for that diagnosis and did not understand why multiple GPs had told me otherwise. I've had other doctors who just agree to disagree and sometimes they refuse to do anything and sometimes they support with me it even when they disagree I had that with postponing the syntocinon injection with my youngest after having had issues with two previous placentas and my reading suggested that as I've contraindications to other hormone medicines that the reasons I'd had such horrible 3rd stages when given it before might be I fall under the vague 'hypersensitivity to the drug' catchall in its contraindications (and because of its use, syntocinon is a hard drug to tell when it's caused an issue or another factor of labour or the placenta has caused an issue). No certain if that idea is correct, I needed a more hands-on managed third stage because of it but it was much faster, with much less pain (my third stage with a previous child was far worse than pushing her out that had been going on moment before), and no retention so I count that as having worked anecdotally. Many things are a balance of risks and benefits with too many unknown factors rather than one right answer that everyone agrees on.
I do think there can be an issue where 'done my own research' means 'saw a post on Facebook', but there is also an issue when people do discuss actual research get dismissed with 'oh, did you google X?' as a way to dismiss people even when discussing interpretations of peer reviewed papers. With the anti-intellectualism, there is both the any source is equal and thinking they can do the research better as well as 'oh you don't agree, so you must just be googling for biased blogs' which can make it kinda hard to discuss these things. It's become all debate rather than any dialogue.
I'm not usually one for adding things to the curriculum as I think there are too many objectives already, but I do think there would be benefits if some of the science curriculum and efforts in science media on how to read academic papers and the values & limits of some of the different methodology. Some teachers are great at this as are some media sources, but I think the curriculum as it stands and even some mainstream media doesn't really help with that understanding.