First, you can certainly verify or disprove the mechanics expertise by his explanation if you have enough expertise yourself to understand it or to understand that he is not using terms correctly.
Irrelevant. If you're an expert yourself then you have no need to trust the experts
Expertise is a spectrum.
Second, causing the car to run does not prove he is an expert. Some problems solve themselves, an overheated coolant system, engine oil that was too cold until it was brought into the shop, a computer that was reset when he blindly unplugged the battery.
Irrelevant. It was just a hypothetical example, you're missing the whole point.
Well if you were trying to make a general point shouldn't you be able to find a specific example that makes it?
Moreover if "results" are defined as "fixing something"
I stated very clearly that expertise is demonstrated by a proven track record of results.
I'll state very clearly that such does not prove expertise it proves effectiveness of some behavior. True expertise results in effectiveness, but effectiveness does not prove true expertise.
Hippocrates helped a lot of people (one can presume) but he was wrong about humors. Was he a medical expert?
Don't strawman this. I know that in our every day life we need to trust the expertise of others, there is too much for one man to know it all; but this is the order of authority:
1.) Reason, there is no substitute, there is no replacement, there is nothing better
2.) Effectiveness (a track record of results)
3.) The esteem of already trusted entities
4.) The esteem of the untrusted general population
2-4 are proxies for (1). In every case whatever power you have to evaluate (1) overrides 2-4. There are indicators of non-expertise and #1 on that list is a refusal to give reasons when asked. All true experts have reasons. Anyone who is effective without reasons is like a mechanic who just got lucky or Hippocrates and his humors. They are not experts and their theories and proclamations are not trustworthy.
You gave an example of someone claiming they are a prophet of God as their explanation for their positive results.
The topic is about whether it is rational to "trust the experts"
No, it's about the fact that there is no mechanism to establish expertise with certainty but to become, to some extent, an expert. That establishing expertise is itself a rational process, the best argument, and therefore in debate and in science (which is a subset of rational epistemology) there is absolutely no place for trust.
Repeat the experiment, don't trust the claim. This isn't an arbitrary rule, it's implied by reason. Science only became distinct from religious philosophy and theology when it eliminated trust from the equation and thereby became a fully rational process.
in your hypothetical the man isn't even claiming to be one.
Of course he is. You simply defined a prophet as "not an expert". A cultural bias that most of humanity did not share up until about five minutes ago.
Yet that was chosen intentionally, how can you tell the difference between a false prophet and an expert when both can produce "good results" consistently?