I can't pretend to know the market, but when I was involved in branded cards, there was a revenue split on the card fees. So then the question becomes "why would a bank want to split the revenue with someone else?". I guess the idea is that they will gain customers they otherwise wouldn't get. And that the partner has some incentive in the card, making the customer using that card for more spend. Ie branded card is meant to increase the market share of the banking partner in terms of CC transactions.

In the case of the card I was involved in, the banking partner actually came to us after a few years saying "We would like to see you make activities to grow number of card holders, otherwise we think the should terminate the partnership". So while we were happy with the card, the bank wasn't unless we put more effort into it and it turned out we chose not to and the partnership was terminated.