Not everyone can be an expert.
Before a witness may testify as an expert, the side offering his testimony must qualify him as an expert, by asking questions designed to show the witness’ expertise - years of experience, training, number of scholarly articles published, and the like.
An expert witness is permitted to offer his opinion on what a certain set of facts implies. This is different from a lay witness, who typically may only testify to what he saw or heard himself. The rationale behind this rule is that the trier of fact - the jury, usually - brings to the fore their own ordinary, common life experience, which they may use to draw reasonable inferences from the facts. But when interpreting the facts requires special expertise, such as DNA, blood splatter analysis, or psychological infirmities, an expert witness can help the fact-finder by offering his interpretations of what the facts imply.
Some fields of endeavor do not lend themselves to different, yet still credible, interpretation, and others do. Two DNA experts may differ as to how to compute probabilities of different people matching a particular specimen, but they would be quibbling over one in a million as opposed to one in ten billion. DNA analysis has certain accepted criteria, and there’s not a useful amount of wiggle room. No matter how much a defense lawyer offers to pay, or how far he searches, he won’t find a credible expert to impeach DNA evidence analysis. (Accusing the police of planting the evidence, of course, might be viable).
But the standard for insanity is a little different. It depends solely on the judgement of the examining psychiartist - did the accused suffer from a mental disease or defect such that, at the time of the act, he was unable to distinguish right from wrong? There’s nothing to weigh or measure here; the expert must simply offer his educated guess. Obviously, certain experts are predisposed to believe that anyone who commits a heinous act likely cannot tell the difference between right and wrong, while others are equally predisposed to believe that it takes a lot to overcome the presumption that all people know right from wrong.
Experts being human, undoubtedly some are swayed in their opinion by the knowledge that they won’t get paid by the defense if they conclude that the accused was sane.
In the end, each of the three possibilities you post has something to do with it. The idea, I think, is that the stronger that facts are, the easier it will be to find an expert whose conclusion is “right” – and the more convincing that expert will be to the jury.
The reality is that money helps - a lot.