That goes against my understanding. Do you have a source for that?
I'm pretty sure credit cards make their money off of interest. It's not from the interchange fees from people who pay their balance each month, as those often go straight back to consumers through rewards programs, and just administration costs generally. The card I use most has no annual fee but 2% cash back on everything.
American Express in particular makes more of their money from transaction fees and membership fees than interest. https://www.investopedia.com/articles/markets/012715/how-ame...
In the US they're around 2%. In Europe they're more like 0.3%, which is why cashback/rewards aren't a thing.
My point is that interchange fees aren't where credit cards make money, it's on the interest of people who don't pay in full.
There is a spread between rewards provided and actual fees (fees are closer to 2.5-4%, rewards usually top out around ~2%)
According to the BNPL industry, fees and interest make up a small portion of their overall revenues. Whether that's wholly accurate is another story.
In sum, they would indeed exist. Its more a question of how large and profitable they would be.
Check out Klarna's financial reporting to get a sense of this: https://www.klarna.com/international/press/klarna-reports-fo...
It may also be irrelevant; I'm more interested in their forward-looking forecasts than their current revenue sources.
Here's a simple test – if everyone paid their balances on time and in full, credit card companies would still survive. BNPL would not.