I have only one credit card to my name - the one I pay off each month. I use it for purchases at shops, gasoline, and to make internet purchases (either direct to credit card or via Paypal to credit card).
One aspect not mentioned here (I think) is that credit card companies also charge the stores about 3% of the purchase price. A lot of gas stations have slim margins, maybe 10%, so a 3% hit from the credit card company significantly reduces their profits. So a lot of lower cost gas stations don't accept credit cards, or post two prices: one for cash, one for credit. Most of the cash-only stations will accept debit cards, but usually with a $.50 to $1.00 surcharge. As a result I've started carrying enough cash to pay for gas, or I drive an extra mile or so and buy the gas at a station that doesn't surcharge for debit cards, and also has just about the lowest price per gallon in town.
So if everyone paid off their credit card bill each month it wouldn't mean zero income for the credit card companies - just a lot less income.
If my credit card company started charging me a monthly fee just to have the card, I'd probably drop it - despite the high credit limit (I have no plans to buy a $40,000 car with it anyway).
One needs to think about the total cost of credit. Let's say you carry a $1,000 balance on your credit card every month, you make the minimum payment plus enough to keep the balance around $1,000. Let's say you have a relatively good interest rate of 12%. That $1,000 you carry will cost you $120 at the end of the year. If you miss a couple of payments, your rate could jump up to 20% or more. Then that credit balance will cost you $200/year. I dunno, I would rather put that $200 into an interest bearing account, or pay off my mortgage a little earlier. Speaking of which, if one must use credit, a home equity line of credit is a much better deal - plus I think you can deduct the interest on that type of loan. Another way is to refinance with cash out... usually even a lower rate of interest and guaranteed tax deduction. Of course, since my goal is to become mortgage/debt free, I don't plan on using any of those strategies. But for someone who is carrying a large credit card balance it's a possible way to reduce the total cost of the debt.
It's also interesting that our banking system is completely dependent upon consumer debt. And the banking system controls the creation of fiat money through debt. Thus, if everyone became debt free then there would be a huge implosion of the money supply. How the heck did we get into this absurd situation? (rhetorical question, I know the answer).