Public intellectuals have a habit of presenting a false consensus. This is not a dig at the kinds of people who become public intellectuals; it's a problem with the way public intellectualism works.
Allow me to incriminate myself...
If I was to lecture you on the topic of moral philosophy for an hour then you may very well come away with the impression that moral rights, both natural and constructivist, are ridiculous. If however, you put an equally skilled defender of moral rights next to me you might come away with a somewhat more nuanced opinion. At the very least you wouldn't be led to believe that only fools and An-Caps trust in moral rights.
Managed by an intermediatary, any relationship between information and the public is open to abuse. Sometimes it can manifest as overt abuses that result from the intermediary refusing to acknowledge an argument, position or fact or sometimes as the more casual failure to give due credence to one's opponents. Such a problem is endemic to our culture of public intellectualism and is a good reason for seeking alternative methods of public education.
In my perfect world, universities, schools and the media would work to make the term "public intellectual" obsolete . To that effect, the training of academics in press relations and making more journals accessible to more people would be great steps forwards (though the latter probably isn't going to happen anytime soon). Offering young people educations that properly prepare them for participating in an intellectual culture would also be a fantastic measure to take. While modern educators are pushing for more emphasis on self-reliant learning and critical thinking, we need to cover a more academically practical curriculum.
Teaching young people not just about the subject matter that they are studying but also about how our knowledge of that subject matter is created and disseminated is vital to building an independent academic culture. If one wants to build a society where everyone can benefit from research, watch new developments and foster interests of their own then schools need to teach students these kinds of skillsets. Our educators must provide young people with knowledge about things like journal articles, peer-reviewal and meta-studies; even if people are not at a level where they can comperehend the content they should at least understand how it is produced. We don't expect a child to understand agriculture but we do teach them about where the chicken drumsticks they love come from: why can't we doing the same for science, economics and philosophy?
In closing I will admit that we do already have a rich and varied culture of public intellectuals. There are academics of every stripe bringing the most exciting and important aspects of their respective fields directly to our coffee tables. There is nothing ignoble about that. Indeed there are so many public intellectuals that if you don't like whoever is in vogue then with a little bit of digging you can easily find someone else. The problem is that for the most part, in my experience at least, people don't go digging; people largely engage with academic subjects as passive receptacles and wait to be filled in by public intellectuals. The problem of popularity and its obstruction of the quest for knowledge is nothing new but its effects can be mitigated by encouraging others to take up a more active and intimate relationship with the subjects that interest them. Until we have a pedagogical revolution in the field of public education I suppose the best we can do is recommend books and articles that are not on the best sellers list and diversify our debates. Let's just try not to be too smug when we recommend authors that people have "probably never heard of".