I am very sorry to hear about your parents. My condolences.
I certainly understand your worries when there is a family history.
This comes with a heaping spoonful o' YMMV, but here is my experience:
Some screenings are allowed under the KK, with age and frequency restrictions. Gynecologists are generally good about preventative screening. For other cancers, talk to your GP, but be prepared for push-back.
Some GPs are more oriented towards prevention than others. My experience has been that outside of gynecology preventative screening has not been considered as important here. One GP frankly said it was a matter of cost reduction - which seems wholly counter-intuitive to me, but there ya go...
The GPs I have seen here also do not seem put as much importance in family history as a risk factor.
Even with clear symptoms and family history, I had to go back to the US to get diagnosed. That hurt the pocketbook, to be sure.
I seem to remember another thread where someone suggested going to France for typical cancer screenings, as attitudes tend to be different than in Switzerland, and costs certainly lower than the US... but I can't seem to find that thread now. Maybe someone will chime in with that info.
If you do go elsewhere, and if doG forbid there is a problem found, ask the doctor there if he or she would write a referral letter that you can send yourself to an oncologist here, just in case your GP is still reluctant to refer you for treatment. And of course bring all your test results, including scans, back with you.
If your GP agrees that a specific screening is warranted in your case, you might mention that you are willing to pay out of pocket if necessary. GPs are under pressure to reduce costs, so taking that issue out of the picture might make a difference.
I keep mentioning GPs, even though many tests are done by a specialist, because you often need a GP referral to see the specialist. This has nothing to do with insurance model. Even with traditional insurance that allows you to self schedule IME getting in to see a specialist without a GP referral has been nigh on impossible. Many require a GP referral in order to manage their patient load.
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As above, YMMV. I'm sure other posters have had different experiences with their doctors here.
Wishing you all the best.