“Trust the science” has always bothered me for two reasons: 1) science is frequently not black and white and anyone who has done hard science research knows there are plenty of competing opinions among scientists and 2) while scientific facts are facts, we still need to decide on how to act on those facts and that decision making process is most certainly political and subjective in nature.
The essence of science is the use of scientific method which have specific meaning and way of doing things. It relies on evidence based knowledge not on any distrust. It does not have to do with authority but you would question if your tools you are using is good (calibrated and not interfering with measurements in an unaccounted way ..etc) or if your methodology is flawed.
Or do they really mean "trust the scientists"?
If I would use it personally I will probably use it to mean trust the evidence based knowledge that the scientific community is using.
Where can one find this knowledge? Are you suggesting regular folk go out and review the literature themselves (most of which is paywalled)? And even if they did and were able to understand the contents, they'd still lack the required context to weigh contradicting results, dismiss old studies now known to be wrong, etc etc.
And that's why "trust the science" ends up being an appeal to authority.
I'm not saying I have a better alternative than the scientific method, I'm just pointing out that the "scientific consensus" isn't some magical spark that is immediately obvious when one reads the literature, it's something that evolves over many decades of research, conferences, etc. And that's assuming there is a consensus for a given topic at a given time. And I'm not even going to get into why reasonably questioning the scientific consensus is a good thing (otherwise it stops being science).