I think to solve this problem, Professional Lawyers use expensive, privately owned/operated third party services like Westlaw and Lexus-Nexus? I think those platforms provide a universal search.
But as an interested member of the public, I can't afford access to those services just so I can research a case to satisfy my curiosity or to correct misinformation that my relatives have seen online and accepted without any further research. I could research the case if I could find the docket, to try to find out what's actually going on (because, very often, right wing social media will make a case go viral, and nobody else is reporting on it, and if you do manage to find anything about the case, you often discover that right wing social media has reported as fact the allegations of just one side in the case/suit, and that reality is a lot messier and more complex and nuanced than the memes report).