No, you aren't making it more costly for the bank. Banks would be stupid to offer such a setup. The banks pass that cost on to the merchants, who have no choice in the matter.
I sure am; almost all of my spending earns a rewards percentage that exceeds the rate charged to the merchant. (This is not even counting the significant sign-up bonuses.)
Anyone (with good credit) can do the same with a modest amount of research.
https://merchantcostconsulting.com/lower-credit-card-process...
No, you are. Think about it, you as an individual can't change how this system works. But if you use a card with 0% rewards, you're gifting that extra 1-2% to the credit card company. If you use a 2% back card, you're taking that 2% that's on offer away from the bank.
Of course they're still making money, the fees are higher. But they make a tiny bit less if you use the rewards card. So might as well.