There was a recent interesting discussion about screening tests on FWR and, iirc, an information leaflet that was targeted at Travellers.
Somebody who works with Travellers explained that although the leaflet seemed short on specifics and might not even have used the word cancer, it had been workshopped to be appropriate and was culturally competent.
I don't know how, but I wonder if the population health part of the healthcare system will need to adopt a, "Are you having your regular screening tests? Talk to your health provider."
That doesn't solve acute or emergency issues. Nor does it address dissonance that is so strong, the person at the heart of it can't contemplate even discussing appropriate screening (e.g., breast, cervical, prostate, reporting birth sex to help interpret kidney clearance results etc.) but there has to be a better method.