It wasn't that long ago that some crackpot tried to convince me that the modern atmosphere had damaged my lungs, so to replace the oxygen I was missing I needed to buy hydrogen peroxide injections. The claim as a whole either needs to have major structural flaws or overturn sizable portions of modern chemistry and physics to have a chance at plausibility, and my biases strongly suggest that instead somebody was trying to skirt legal oversight in order to con me.
It's also got to be a matter of sufficient consequence to be worth the time.
Extraordinary claims require extraordinary evidence, etc.
That is why it is important to support your local crackpot. Much as staid English society always valued their eccentrics. The more ultrarational the person, the more important for that person to spend quality time at the fringes.
There's also the claim that someone is being "close-minded", when they reject an idea they have soundly researched and therefore need not waste any more time on. As Harlan Ellison so eloquently put it: "I'd gotten all the literature I could handle on the subject from a certain Thomas Aquinas" in regards to the god hypothesis. More often than not, "close-minded" is a conversation stopper, designed to deflect from the accuser's very own ignorance and credulity.
Same thing goes for creationism, AGW denial, anti-vaxx, anti-mask, the QAnon cult, trickle-down economics, etc.
I find that breaking the habit of being quick to judge leads to open-mindedness. This is also a tenet of Cognitive Behavioral Therapy - your first thought and subsequent feelings that arise from that thought are often wrong.
but, in my experience, most people that put "I'm open minded" in a self description are signaling they believe in certain specific policies and they are actually close minded about whether those policies are good or bad. They've already decided they are good and won't consider (have an open mind) to any suggestions otherwise.
The former may be useful. The latter is usually just annoying.
Of course, I agree with you that open-mindedness is important when dealing with intellectually honest people.
Lately, in response to the relatively intense polarization of beliefs, I've been contemplating open-mindedness, and whether discussions starting with establishing an open-minded context could get people talking again.
To me, open-mindedness means that you're receptive to new ideas and experiences and generally do not reject them outright. When presented with implausible or seemingly impossible information, an open-minded person will listen to the basis of ideas, concepts and information and weigh that basis before making a determination about whether or not the idea is true and worth assimilating. Or they may try a new experience with less pre-judgment of whether they'll appreciate it.
In my view, there is a critical nuance in open-mindedness: It applies whether or not the idea is consistent with what I already believe to be true. As an open-minded person, I feel that I need to regularly reevaluate my beliefs as new information comes to light, even if that new information directly contradicts a basis for a firmly held belief.
I'm curious to know examples that you've had where you've learned new information that contradicted what you already firmly believed, and through open-mindedness, changed your belief accordingly.