Why do banks pay a compound interest on deposits instead of simple interest?

In a sense, banks don't pay compound interest, they just pay interest at the end of a certain period. If you leave your original deposit and the interest they paid for that period in place for another period, then the pay interest on both the original deposit and last period's interest. This way, the interest begins to compound: in the second period, you get interest on the interest; the period after, you'll get interest on the interest on the interest etc.

(Things are slightly complicated if they, say, calculate interest on the daily balance, and only pay it monthly, but that's just "detail").

Also, as chepner notes in a comment, market forces will tend to mean they will end up paying "the same amount of money" either way, whether it's calculated as simple interest at one rate, or compound interest at a different rate.

Compound interest encourages longer-term investments.

Banks want to hang onto as much of your money for as long as they can. With simple interest, you get the exact same interest payout every day/month/year - it doesn't matter how long the money has been invested. After putting some money in a bank account with simple interest, I have the same incentive to keep it there after 1 year, or 5 years, or 10 years. I can withdraw the money, and reinvest it a year later, and lose nothing more than that single year's value in interest.

With compound interest, your interest payout goes up with each passing day/month/year, as your principal compounds and the value of the interest goes up. Because of this, my incentive to keep my money in the bank is constantly going up. I stand to earn more interest in year 2 than I did in year 1, so I'm effectively incentivized to keep my money in the bank for as long as possible, in a way that simple interest does not. If I withdraw money for a year, I'm missing out on that year's interest, plus all the future interest I would have earned on that year's interest.

Simple interest and compound interest rates could be set such that they have an equivalent payout, but only at a certain point in time. Before that break-even point, simple interest would pay more, and after that point, compound interest would pay more. A savvy investor could invest in a higher-yield simple interest account, and simply close it and reinvest or switch to a compound interest account that would have a higher long-term yield. Simple interest doesn't expressly prevent one from taking advantage of increasing principal, but it puts an onus on the consumer to constantly close/open accounts or expressly reinvest interest as principal.

Overall, compound interest is easier for the consumer since they don't have to muck around with their account to maximize interest, easier for the bank since the bookkeeping is simpler, provides a potentially larger long-term benefit to the consumer, and provides a means for the bank to incentivize long-term investments.

You can get your scenario in any bank: invest in a certificate of deposit, and on maturity, roll over the CD into a checking account that doesn't qualify for interest.

Sure, to your leading question, a bank would be happy to keep you as a customer if you leave the money in the non interest bearing account. But they can't compel you to leave the funds there (where they can keep you in the CD until maturity with forfeit interest or fees).

If your stated plan is to roll over the proceeds of the CD including interest into another one, or the same setup at the next bank down the street, then let me tell you this exciting news: banks have savings accounts that do this for you, on the granularity of 1 day than one CD term. The interest rate on each day may vary as a function of the federal funds rate or marketing decisions (where the rate set for the CD won't change over the term) but it automatically rolls over, and compounds daily.