Disclaimer: I'm a physicist but have no expertise in solid state physics. I remember when the whole high-Tc thing started. My recollection is that the discovery broke what was at the time considered to be the mainstream understanding of superconductivity. And, a more comprehensive and satisfactory theory has not yet emerged. As a result, we're in a sort of tortoise-and-hare race between theory and experiment, where each one advances a bit when the other one catches up and makes a new discovery.

A fundamental reason is that with current hardware and software large quantum mechanical systems are hard to simulate.

So we can't just through a computer at it, to try out all possibilities.