Could there be an unconscious psychological aspect to our apparent obsession? Not anything psychologically defective, but rather I’m thinking of instinct and intuition and gut-feelings; in the unconscious assimilation of information which is processed by the brain and revealed to the conscious mind through a variety of ways. Instinct is a genetic inheritance which can have a psychological ingredient that is generally thought to enhance our ability to cope with vital environmental contingencies, but it does not necessarily lead to any intellectual solution which we might seek. But we often say that we know instinctively that Bessler’s wheel was genuine.
Intuition is a person's capacity to obtain or have direct knowledge and/or immediate insight, without observation or reason. It's the "gut feeling" you get. People often place an enormous amount of faith on their intuition even making decisions that seem to go against all available evidence. I prefer the term ‘insight’. Insight is the capacity to gain an accurate and deep understanding of someone or something.
So we who labour long and intensely in the search for the solution to Bessler's wheel know intuitively that those who would have us believe that we are wasting our time, are wrong, so very wrong that we simply cannot wait to prove it. In this perpetual pursuit perseverance is paramount.