An intelligent person would trust institutions that change their mind more, because it's the right thing to do.
Institutions that remain utterly consistent in the face of evolving knowledge are the ones you should be wary of.
I saw an analogous thing in the much simpler world of signal processing and statistical estimation. Estimators and control systems sometimes oscillate as new data is acquired. It seems counterintuitive, more data should just improve accuracy, right? As if to converge toward an underlying true value? No, sometimes the best possible estimate and control output oscillates gently as more data is acquired, without any inconsistency. It took me a while to appreciate that.
Back to the big picture of institutions. Inconsistency over time may seem dissonant, but it shouldn't. It can be the most correct and accurate recommendation over time as new data and knowledge is accumulated. In an evolving situation, you should be seeing this.
The fact is people do cite inconsistency as a reason to disbelieve, and it puts public health policymakers in a dilemma. If they tell the truth and give the best advice to follow, people don't believe it because the truth is complicated and counterintuitive, and best advice rightly changes over time, and in different locations, circumstances, etc. So they have to walk a line between fully detailed truth, and simplified advice that people en masse are more inclined to believe and follow.
(I don't disagree with your other assertions about accountability etc).