The EU Interchange fee antitrust investigation, a result of which Visa and Mastercard have offered to cap at around 0.2%-0.3% of transaction value across the EEA (previously interchange was levied at different rates across different countries in the EEA, but now there's "Domestic", "EEA", and "Rest of the World" rates. However, interchange is really only Visa and MasterCard's cut - as in, the cut the actual payment network. Processing fees for cards are still levied on top by payment processors and then distributed to issuers (say, 1-2% depending on whether it's a normal Visa, or a corporate Visa, or a MasterCard Signia, or normal MasterCard or it's done over the phone, or if you are an airline etc.) and it's that fee that issuers take and use to reward you with points they buy from loyalty programs. Debit cards, credit cards, charge products - all of them have different rates, but the only ones that consistently give enough margin for decent rewards (i.e. 1 point per USD/GBP/AUD etc) are credit cards, specifically premium ones. Some co-branded cards, such as Airline co-brands and Hotel co-brands give the co-branded merchant an interchange/processing rebate for spend at that location (e.g. a Marriott hotel accepting payment on a Marriott credit card gets a small rebate on processing costs for that card usage - which usually just goes to the hotel buying you more reward points).

Ironically, everyone thinks that AMEX is the most expensive, but I've seen total processing fees for Visa Infinite and a MasterCard World Elite surpass AMEX charges...