@kmd - Caveat: I have little experience in Switzerland, but...
To help (anyone) troubleshoot) this, it'll be useful to provide more details, so various possibilities can be eliminated.
One such possibility is something fairly obscure called "3-D Secure" which is (or was) a system used by some payment processors in Europe. You can read more about here in this old thread:
3-D Secure credit card payments - wha???.
This is a post from 6 years ago, so some things may have changed, but based on the fact that your payment is simply failing, yet your bank insists they have not declined any charges, it's worth looking at this ("3-D Secure") as a potential source of the problem - because that is exactly how it looks to you (and your bank) when 3-D Secure has short-circuited your transaction.
Short version: it's a security protocol that's fairly widespread in Europe which requires a one-time security PIN to complete the transaction. But many (most?) credit cards issued by American banks are not set up to use it, so transactions just fail with no useful feedback. It looks to the user that the payment has just failed (which we often interpret as a "decline" due to suspected fraud), but it has NOT been declined by your bank - the payment request never made it to your bank, they never saw it, rather, the payment was "short-circuited" (and dropped) by the vendor's payment processor.
This used to trip-up foreigners regularly, but seems to have diminished in recent years as some US banks have made their systems more tolerant of the protocol, but is still not widely understood by foreign consumers (and most front-line US-based bank employees) and it could be happening to you. Important: Typically it only interferes with online payments, and in-person payments at automated (unattended) systems, eg some gas stations, train ticket machines, highway toll booths, etc. With payment terminals staffed by a live human, it does not seem to be an issue, so there's a distinction to look for.
So, to the troubleshooting question: If you look back at the places where your cards failed, were they all at automated, UN-attended systems (or online purchases)? Or were some of those payment failures at a human-staffed payment location?
If the failures all were at automated terminals (or online), and the same cards have worked without issues with a human-staffed terminal, then that potentially points to the "3-D Secure" system. If the failed transactions were all at human-staffed terminals, then never mind about all this 3-D Secure stuff.
Hope that helps, good luck.