Believing in science is a very old thing, Maya have believed in their eclipse prediction science so much that were even making human sacrifice to save the sun from being eaten.
We should teach the public not to "believe in science", but the main skill of any scientist: to try to find errors in the theory you think is right with more ardour than in the theory you think is wrong.
I actually think this was one failure in our response to the pandemic. Take N95 masks or testing or what have you. There’s been so much back and forth on how well things work and changes of messaging - often driven by real studies, thus reflecting the uncertainty we’re talking about! - and I think that’s the cause of some distrust you’re talking about. In an alternate reality CDC would came out on day one to say “N95s are obviously good, and so are widely available rapid tests so let’s make a ton of them” and stay on this message. It’s the uncertainty and doubt that creates distrust, not the other way around.
The initial stance on masks was to lie that they are useless so that people don't buy masks that were needed for doctors, so if CDC was to stay on message it would stay on that message not the one that masks are good.
Your proposed approach of a few people making decisions and the rest not even being allowed to debate, can not create trust, it can only create USSR or North Korea.
A sidenote on CDC, the null hypothesis is institutional inertia: there were no mask mandates during many recent pandemics so CDC was simply following the long standing policy. This doesn’t require lying: if you believe that a policy is useless and will actively harm supply of masks it makes sense to come out strongly against it. I’m open to evidence to the contrary though.