Leaving aside common EHR / central database being orthogonal to universal healthcare, as addressed in sibling comments, having this data centrally still doesn't even make this as easy as you hope.

'patient complains of increased nosebleeds' isn't structured data you can query (or feed to ML) like that. It actually takes a physician having this kind of hypothesis, to then trawl through the records, reading unstructured notes, creating their own database for the purpose - you know, had/did not have nosebleed, developed/did not develop pancreatic cancer within 4 years, or whatever - so then they can do the actual analysis on the extracted data.

Where I think LLMs could indeed be very helpful is in this data collection phase: this is the structured data I want, this is the pile of notes, go. (Then you check some small percentage of them and if they're correct assume the rest are too. There's already huge scope for human error here, so this seems acceptable.)