Belief in fake searches is positively correlated with number of rejections and years on the market.
No, understanding of the existence of fake searches is positively correlated with number of years on the market.
I've been in at least three, two very early on, which I didn't understand were fake. With that experience, I did see the third fake search coming but I went to the teaching demo anyway for the experience (I wanted to brush up skills that year) and to learn about them. The offer went to the internal candidate it was obviously written for - as had the others - but I took another job without regret, having learned about them.
IME, the tell-tale signs are a job description written with extreme specificity, often matching the background of one of their adjuncts or post-docs, and they state a policy of not reimbursing for campus visits.