Few tricks.

1. Card services actually get a small fee per transaction from the retailer so they just get paid every time you use their card which can add up.
2. If you fail to pay off the balance for any reason they can charge generally a 20%+ compounding interest that for most people is probably closer to 27-30% and quickly replaces the minimal cash back.
3. Spending habits and ability to sell the data to other card/loan providers and advertisers can help generate additional revenue. Think maybe you consider them trustworthy and they offer a car loan for $30,000 and you take it happily without looking around to find out they have a much higher interest rate on it and they get an extra couple grand from you.

Interest rates if you don't pay off the full amount

making the shops pay more to use the payment service, so that the shops then increase the prices, so that you pay the same as before

Just nitpicking because I enjoy these thoughts:

When the shop increases prices, it has to do it for all the customers, including the ones without credit card. So a part of the cost is offloaded to other types of customers. While credit card customers should see a slight increase in price, it should not be as much as they saved previously. So still a net win for them, at the cost of others.

As others pointed out, the real scheme is probably entirely different.

Credit card companies make money off selling your spending habits to information brokers.

The business model is depending on card users accumulating interest. So they give you perks to get you to use the card and enough people won't pay it off every month that they'll make all their money back and more.