But cards are offline from the perspective of the consumer. Sometimes even on the merchant side of things. Not that it is an important distinction nowadays--but I have definitely tried to pay with a merchant's own app-based payment solution that refused to load due to a bad cellular connection. I haven't looked into how Wero will handle this.
Most payment terminal nowadays use 4g network and it is not uncommon to see shop/restaurants employees in some areas trying to desperately get a signal by moving the terminal close to the door or window.
They need much much less data than your phone. They could process several transactions with less data it would take for your phone to load the HTML of the payment page, let alone the Javascript or the bank's logo.
Also, such terminals often use multi-carrier data plans that can use the best carrier available, while your own phone is stuck with one of the options (of course, you always have the worst one).