Actually, it’s not “cheaper” for the banks that their customers use their debit cards. It’s because they get their own “skim” from every single transaction. The business it is used at is charged around 2% for every debit card and credit card transaction. That percentage goes to the bank. It is in their interest that EVERYONE use a card for EVERYTHING.
MC, VI, AMEX charge a fee for their service and is paid by the company accepting your charge or debit card. The fees vary, but can be around 2% but can be higher or lower depending on the companies volume of transactions. AMEX can be as high as 6%.
There are companies you have never heard of that process/facilitate the transactions for the credit card companies/banks. It isn't cheap to process - takes an enormous amount of capacity for the tech equipment and IT labor. That almost instant approval of your card at the checkout isn't cheap.
Banks charge companies when depositing checks and cash, too.
Nothing's free and besides companies DO prefer the use of credit/debit cards because they don't have the internal control problems of handling cash.