When I think about experts, and their role in society, I try not to use the obvious examples like scientists telling us about global warming, or economists telling us about global free trade, or even statisticians telling coaches about the alleged fallacy of the "hot hand". It is better to use an example closer to home, which we encounter more often, and see the results of on a shorter time scale. I think one of the best examples (for me), is car mechanics.
If you are a car person, who knows a lot about working on your own car, then this won't be a good example for you. Likewise, if you live in a dense urban area with great public transit, and you don't have to own a car in order to get around, then this also may not work for you. But I live in an area where it is a handicap to live without a car entirely, and yet I don't know a lot about the details of how a car works. I depend on it working, but don't know much about it.
I need an expert on cars to keep my car functioning, but I don't know enough about cars to be able to tell if they are being straight with me. When they say I need to pay them $2,000 for a new transmission, is it true, or do they just believe this because they need the money? Relations between me and my mechanic rely upon trust, that I believe they are taking my welfare into account when making their decisions. Of course they will also take their own welfare into account, because they need to pay themselves for their labor and not just the parts.
If I begin to think that they are not making decisions with my welfare in mind, then I will be hesitant to take the car in when it needs repair. I might just try to fix it myself, or even decide to ignore the weird noises it is making. This is understandable, even predictable, if I live in a society where car mechanics cannot be trusted to take my welfare into account, but that doesn't mean it will work.
If car repair was done by an institution of car mechanics, all together and removed from personal contact with their customers, then this trust is more likely to erode. The car mechanics may even think that they are taking my welfare into account, but without face to face contact they will tend to think more about how to make the car perfect, and make themselves wealthier into the bargain, and less about the fact that they are taking all of the money that's left in my bank account. Eventually, they will face a revolt of their customers, who start taking their cars to their unqualified car enthusiast neighbors for repairs.
Society needs people like economists, sociologists, climate scientists, doctors, etc. to work on ideas for how to make society work. They need to have trust that the people plotting the course for our society have their welfare in mind. But, because academics and the working class are rarely in face to face contact nowadays, that trust does not exist. This doesn't mean we have no need for a coherent economic policy or a coherent plan for dealing with social problems such as drug addiction, suicide, racial tensions, etc. It just means that the people who ought to be in charge of formulating such policy, have no mandate to do so from the larger society, and have little understanding of (and less respect for) the subculture of large portions of the society that pays for their academic careers.
There are thus a couple of problems with the assertion that we have to trust the experts, and not be "anti-science" or whatever equivalent term fits that topic. The first, is that sometimes the experts are wrong, and they're wrong more often when they are in a position where they only talk to, or at, the non-experts, and don't listen. The reason that I know, if my mechanic tells me that I need to spend $2,000 on my car, that it is really true, is all of those times where he said something like, "I changed the $0.50 part that was your problem, you don't need to pay me anything". Yes, that really happened with my car mechanic. When he is looking at my ancient car, he knows that really every part of it should be replaced, but he also knows that his reputation, the trust that his customers have in him taking their interests into account, is his most important asset. So, he will tell me which parts need to be replaced right now, which can wait until next month, and which can be put off.
He also probably is more likely to take my interests into account, because he talks to me face to face. If he were living in an ivory tower of all car mechanics, and malfunctioning cars were sent there without their owners near at hand, and he rarely had face to face contact with his customers, he might (with the best of intentions) say that everything which is not top quality needs to be replaced. If I have to have a car in order to get to work, I might grit my teeth and do it, and struggle for months to pay it off. But I would be easy prey for any non-qualified mechanic who wanted to offer me his repair services instead.
Moreover, if he and I never had face-to-face contact, or if all car mechanics had political and ethical values which were out of sync with the society at large, then car mechanics in general would start to be a polarizing influence. They might say that every part of my old car had to be replaced because otherwise it would leak oil sometimes, which would be an environmental disaster. Or, they might say that I needed to replace any part which might have been damaged that one time I was late in changing the oil, because otherwise there would be a safety hazard, and safety is not something you can compromise on. By thinking only about the car, and not about the person, they would no longer have trust from the people who are, ultimately, paying for them.
The second reason, of course, is that sometimes experts actually are wrong (see the link on the "hot hand" fallacy above, for one such possible case). Doctors scoffed at the idea of washing their hands between delivering babies, at a time when midwives did. The IMF and World Bank have given horrible advice to Third World countries about how they should develop their economies, and it has often been observed that the countries which do the opposite of what those experts advise, seem to do at least as well as those who take their advice. Being an expert means, in any field with a reasonable level of development, that you will be right more often than someone who is not an expert. However, every field has examples where experts "knew" that the common opinion of those with experience in a field was wrong, and it was the experts who were wrong instead. If those experts and the non-experts they are talking to share the same values, generally, and they are part of the same society, then the feedback works both ways. People are more willing to listen, if they are listened to.
In the case of car mechanics, this is not an issue. Car mechanics live among the rest of the society, and by and large share the same values on most issues. If you find a good one, you can trust them. Since they don't all live in the same car-mechanic town, and they don't drive out of the profession everyone who doesn't think like them, you can probably find one that you can trust.
But, what if experts don't live amidst the rest of the society? What if they live in a bubble, in a highly concentrated, car-mechanics-only town where they get to talk about cars all day with other experts? What if they all come from a particular part of the political spectrum, and start to drive out of the field anyone who comes from a different part of the political spectrum?
Just because you can't trust the experts to share your values, or take your interests into account when making their expert decisions, doesn't mean you don't need their expertise. But just because people need their expertise, doesn't mean they will trust them enough to follow it. What would the streets look like, in a society where people didn't trust their car mechanics? There would be broken down cars all over the place, maybe others crawling along with their flashers on at half speed, and the occasional catastrophic failure resulting in deaths or at least traffic snarls. Trust is important. I fear that the way our experts live in a society apart, is corrosive to it.