I can't speak for Cuban, but I was describing taking the 20-40 regular blood tests plus whatever someone may be interested in for more personal reasons, more regularly. Not necessarily adding breadth to the data, but regularity. Yes, a genetic test that comes back negative is unlikely to change.
Of course, we think it won't change based on a belief about how genetics works that isn't frequently tested, and poorly researched. Some genetic changes that can happen during a lifetime weren't really accepted until 2008, and we don't have a good understanding of that impact yet. Another example I can give you is Chimerism -- the possibility that someone has two sets of DNA working in their body, sometimes with different chromosome types. This is the sort of thing that tends to be diagnosed by accident, sometimes during transplant typing. It simply wouldn't be uncovered during a single genetic test, but multiple tests over time would make it readily apparently.
I think that's a bit extreme, though... simpler examples are borderline cases of routine blood work. The de-facto standard is to compare blood work to general standards, but we don't adjust for variation in individuals. A white blood cell count of 11 is borderline, but if I came to a doctor with symptoms and an 11 WBC, an might be prescribed, since "normal" is just below that. If, on the other hand, I had years of "healthy" measurements where my WBC was normally 10-11, we'd realize that I had a naturally high count and probably conclude that this symptom may be something that is not affecting my WBC. Or, years of high tests might be a symptom of something else, but still different than a single spike from a person with a proven healthy average of 4-5. If I only ever have the test performed when I'm sick then all you can compare me to is the standard population. That's bad statistics, bad science, and it ought to be bad medicine.
Now, I've also heard from doctors that said they wouldn't treat this patient differently because it goes against their training. So, yes, I agree the US has a ways to go, and I agree we need a culture change. I don't think the way to make that change is to shoot down people who are willing to be a part of it and promote it. I do think that even if you dislike the precise mechanism Cuban recommended, it still sounds like a step towards the more openness that it seems we agree is superior in your Scandinavian example, even if the path isn't direct. We won't change overnight, but we won't change at all if we stop people from trying.