This doesn't really happen much. First off, only a tiny percentage of people are thinking about this critically and independently. So it's more like "People are told X is bullshit, and choose to believe it". That holds regardless of the truth value of X, and has more to do with the intentional undermining of science as a source of finding truth.
So yes, we should listen to the scientists. Not the people trying to spin what they say, but the scientists. And not individual scientists, but disciplines as a whole. We should be able to understand when there is meaningful consensus and when there is not. How confident the community is in quality of the work. What ways there would be to interrogate it better. We should be funding competent and trained scientific journalists to synthesize and present this things in as objective a way as they can, instead of listening to PR hacks and politicians spin a random paper they found to support the conclusion they want.
Absolutely there are problems in science, and to some degree we have caused them by creating poor incentive structures. So lets fix that. We should probably be focusing more on evidence-base policy, and creating oversight structures that keep it working well.
After all, for the questions that science is capable of addressing, it is far and away the most effective approach humans have ever come up with. And as recent history shows us yet again, if you throw out those voices they will most likely be replaced by blind true-believers and opportunistic mountebanks. You can't honestly believe this will improve things, can you?
Of course, we shouldn't only listen to the scientists. But these day's we've swung too far in the other direction.
Within the context of Brexit, the leavers had their own group of "experts" i.e. the ERG but any attempt at economic prediction was clearly shunned on the grounds of it being "Project Fear" despite no specific biases being demonstrated. Because that's the only way, along with blatant lies, to get people believe that leaving the largest trading block - possibly even with no replacement deal with anyone - could possibly be a good thing (At all costs being the current mantra i.e. No Deal)
It's not the lack of listening to certain experts that bothers me, it's them clearing playing the card. Gove, and most of the others on the leave side, is very well educated and damn well knows that careful thought ("expert"ism) is important.
It's very interesting to me, cause I observe experts in expertise everywhere in Russian media, and have been feeling lately, that they'll close themselves in no time, if they proceed that way, or turn into buzzfeeds.
Btw, I think this is equally true of research outside the social sciences...but there is just far more scope to engage in puffery within the social sciences and, unfortunately, people are often far too lazy to counteract it.
It isn't even about data or the distinction between hard and soft...if you can make a solid argument that is justified and you have no data (which is sometimes the case) then fine. The issue is that, in most areas, the evidence doesn't justify any clear conclusion. Again, you see this in harder sciences where the title is eye-catching, the data less so (often in proportion to the nonsense in the title).
Areas like "peace building" and "conflict resolution", mentioned in the article, are deeply political areas. In my opinion, they rarely do much to advance human knowledge. For example, as the article inadvertently shows, peace building is explicitly based on the idea that democracies are best because they don't attack each other. Sounds great...but there is no data to support this conclusion.
Note well: This does not excuse purging intellectuals, in Turkey or anywhere else. Intellectuals aren't gods, but they aren't vermin, either.
Speaking of which, I recently saw a petition which was sold along the lines of "100s of scientists sign petition condemning 5G", where the vast majority of signatories were either scientists not working in a related field or physicians (Who are not scientists at all).