European countries such as Sweden and the Netherlands are at the forefront of submitting themselves to a cashless economy, giving control of the payments infrastructure to Visa and MasterCard. So it's certainly not obvious to all Europeans.
The network is owned by the Dutch banks, which also provide the online payment platform iDeal.
The fees are low and fixed. Offline, between 15cts and a couple of cents ( I don't know how low for the big retailers, 5cts ? )
The online fees are a bit higher, starting at 29cts.
There are no variable fees. There is no 'refund' or 'insurance'. Lots of shops do not accept creditcards ( I know of no big retailer which does ).
Note that I don't believe having companies - especially large ones that isn't primarily banks - in control of payments of any kind is a good idea, I think it's a terrible idea.
Yes, us Swedes use cards a lot, and almost every card in Sweden has Visa or MasterCard stamped on it. Most are debit cards, and the well know logos weren't there to begin with. Initially, almost all cards were debit cards issued by the various Swedish banks. Banks that people used to trust back then.
While those payment systems are technically private, they aren't perceived as such. Maybe because of the banks, maybe because they are regulated quite heavily, but people generally have confidence they won't be cut off by the bank. Personally I don't know anyone, or have heard of anyone getting denied a debit card. While that's not a useful statistic, it shows why people might perceive it the way they do.
Since 2017 the laws implementating the then new EU banking directives demands that the banks must offer any basic payment solution/product to everyone if they offer it to anyone. There are some exceptions, mostly regarding money and banking fraud, but it seems to be somewhat rare in practice, at least it doesn't seem to have garnered much attention in news.
There are however talks of introducing some kind of legal tender laws, as it is currently possible to get into a situation where you have money, but can't pay. Which is obviously terrible.