I never even came close to stating that "the people we trust are always right". You appear to be viewing the word "trust" entirely too much as a binary state. Tabooing that word for now...

Since we can't always spend the time and effort to verify a claim, it's important for people (and thus important to teach children) to be able to quickly assign a probability to the likelihood that a person is correct when they say something. There are a number of factors that can go into such a calculation, and they will differ based on the statement and the speaker. A person who has a known history of trying to deceive others should be assigned a lower prior probability of correctness than a person who has not shown such a history. A person who makes a large number of honest mistakes should also be assigned a lower probability than somebody who doesn't. However, that is where the similarities end.

An ardent young-earth creationist's views on geology should probably be assigned a very low prior probability of correctness, but if that same Y.E.C. has a PhD in economics, their views on something like inflation should probably be given a higher probability of correctness than those of most people. When you know the speaker has a tendency towards non-malicious incorrectness in a given area, you can use that information to discount their beliefs in that area without writing off everything the person says in all areas. You should still be skeptical of anything they say that seems unlikely, and you should expend the effort on verifying the claim that is appropriate to your live and the value of your time and effort given the probability that they are correct (taking into account things like how well peer-reviewed the position is, whether it contradicts common sense, etc., but also considering how well the person can be expected to know the field and whether they have any known reason to deceive people about it). All else being equal, there's no reason I know of to have a greater expectation that a Y.E.C. is incorrect about inflation than somebody who is not a Y.E.C.

For a person who has a known history of intentional deceit, it makes sense to use a lower prior probability of correctness for everything they say. A politician's promises are an excellent example; without going into any actual political side, I think we can all agree that politicians are far more likely to make false promises and deceptive claims than the average person who is not in (or striving for) a similar position of popular authority. There is reason to assign a lower prior probability of correctness to almost anything a politician says (publicly) than there is for an otherwise-equal non-politician.

Now step back from the broad categories of things like Y.E.C.s and politicians, and consider the people around you in your daily life. Most of them will have no motive to intentionally deceive you, but some will. Many of them will have biases towards incorrect positions in a lot of areas, but it would be inefficient (and socially awkward) to act as though a friend who has a known bias about a sports team as though they're a pathological liar about non-sports-things just because they're completely blind to that team's quality and you've caught them in a number of false claims about the team that they should have known were false. On the other hand, some people just are unreliable about things, or think it's funny to convince people of lies for no purpose but their own amusement, or have developed a reason to want to hurt you personally and will say whatever they think will have that effect. It is important to be able to tell the difference between those people and those who merely sometimes make honest mistakes.