Downside of this as a tourist/visitor in Denmark at least some smaller shops force you to pay with cash since they don’t accept visa/mastercard but only the national debit card. Just a minor annoyance at least as a Swede since we don’t usually carry cash these days.

I reckon a situation like that is un-European, the Commission will probably step in at some point.

How is it un-European for Denmark to prefer (if they actually do so) a national Danish (hence EU member) card above an American card?

that would be suicidal, but an EU initiative for incentivizing low-fee cashless payments is not out of question.