I know this question is about restaurants in Vienna, but we seem to have veered into Costco and Californian fuel.
So, in the UK, charging extra for a card transaction is illegal, and I believe so in the EU.
Costco in the UK does not issue or partner with a particular card, although at the beginning it was tied to an AMEX card. In the store, and the food court and at the petrol station Visa, MC and Amex are all accepted. The petrol station is all pay by card at the pump; in the store cash is also accepted; in the food court it is contactless card only at a self serve ordering screen, and cash may be paid if you search out the one person who can take it while serving others.
I carry a £10 note on the off chance I will need it during the day, and I carry a physical card for the same reason. I do all my purchases with my phone and although I carry my wallet I never take it out. Never need cash, need a physical card every few months. I don't think I used it in 2024.