I can't answer for Paul Davis, but expertise is often a matter of persuasion and marketing. One might be qualified through credentials or experience, or some combination of the two, but many expert witnesses also lean on their prior service as a witness or training instructor, which is treated as commensurate with experience even though it is actually an assertion of social proof.

There isn't a hard methodological requirement - rather, testimony is accepted (or not) on the basis of adversarial dispute. In some dispute between peers it would probably work as you hope, eg a metallurgist offering an opinion on similarity of two industrial processes.

But consider criminal trials, which often turn on expert testimony. There you have a big asymmetry between the government prosecutors (who come into court with a de facto presumption of integrity) and defendants who may be unsympathetic, as well as reliant upon a poorly-resourced public defender. An officer's suspicions or use of force may often be supported in court by expert testimony more notable for its intensity than its applicability; the jury is encouraged to imagine the worst-case scenario rather than assess the probability that it happens. The quality of forensic evidence and procedures can and does fall short of scientific rigor. Often this involves unscientific claims of perfect reliability which leverage the jury's uninformed respect for scientific matters; a 2009 government report on forensic evidence found that 95% of testimony submitted by the FBI fell short of scientific validity. A subsequent study summarizes efforts to place legal consideration of scientific evidence on a more rigorous footing: https://obamawhitehouse.archives.gov/sites/default/files/mic...

Sometimes expert witnesses are just not qualified. During the last presidential campaign a woman alleged she had been sexually assaulted by Joe Biden when he was a US senator, which naturally attracted journalistic scrutiny. She had previously been an expert witness on sexual assault in 10 criminal cases, based on credentials and experience. Subsequent investigation (first by the media, later by a DA's office) established that she did not hold the academic qualification she claimed to have, though whether this claim was rooted in misunderstanding or mendacity was not established. I don't believe her testimony was dispositive in those trials, but in all 10 her credentials were accepted rather than verified.

A big issue in US law arises out of the adversarial nature of the legal system: even where expert testimony may be invalid or outright false, if it is not successfully challenged at the trial stage then it is very difficult to raise its invalidity as an issue at appeal after conviction/litigation. You could have proof or even an admission of unreliability, and that might be fully accepted by the appeal court, but an unjust conviction might be upheld on the basis that the expert witnesses' expertise or testimony should have been impeached at or before the trial stage, and the defense's failure to do so was a strategic error rather than a failure of the court. This is exacerbated by the fact that appeal courts consider only arguments of law, but not of fact. The overall good of the legal process is given greater weight than the reliability of any individual outcome; in practice, a certain amount of injustice is accepted as the price of getting things done.

Thus, many criminal appeals involve a convict to first attack their former defense counsel for bad lawyering and to win that argument, in order to force reconsideration or a new trial in the original court. Years can pass between the discovery or proof of error (or even innocence), and legal exoneration and release from prison, perhaps even from death row. You can see how the uncritical acceptance or over-valuation of 'expertise' can have a compounding effect as a case moves through the legal system, with many flawed outcomes going unfixed because the burden of repair exceeds the injured party's legal leverage, similar to the accumulation of technical debt.

If you're interested in this topic (and you should be, because the legal system of a country shapes not only case outcomes but policy formation, cultural mores, and cognitive styles), read Adversarial Legalism by Robert Kagan, whose basic ideas are succinctly summarized here: https://law.stanford.edu/publications/the-same-only-differen...