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| But isn't the Corner Card also managed by one of the aforementioned service providers? | |
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No.
The following major credit card *processors* (in order of relevance) are present in the Swiss market:
- Card Center (UBS controlled)
- Swiss Card (CS controlled)
- Viseca (Raiffeisen group controlled)
- Corner (Corner-controlled

)
(I am not sure whether Cembra actually are their own processor.)
Basically, a processor performs all the boring administrative and call center tasks for an issuer (e.g. a bank, or an airline, or ...). So you want your card with an issuer who contracts with a good processor. And do note that issuers may switch processors over the course of time (happens infrequently, though).
Some issuers might also set up their own call center and then make use of the facilities of the processor ... it's all a bit of fun
The best you can do is work through the credit card offerings available to you and assess every single card for your needs.
I myself have a Migros Cumulus credit card, by Cembra Money Bank. But only because a) my life does not depend on it and b) it is about the cheapest thing you can get for CHF as the card currency.