It's only a debt for the business who receives the loan.

When a business receives a loan it shows up as an asset to them in the form of a bank deposit. The business then usually uses that demand deposit to purchase goods and services, so people who don't owe debt to the bank get those deposits in their accounts, and spend the deposits, etc., etc. So effectively, private banks create money.

You missed the part where the business gives the money for those goods and services back to the bank plus interest and the fact that the bank already had the money to give, nothing was created

The bank doesn't usually "have the money to give" when it makes a loan.

Let's say Bank A loans $1000 to a customer. It creates a $1000 bank deposit in that customer's account. On the balance sheet it looks like this:

Bank A:

(Asset) Loan to customer of $1000

(Liability) Bank deposit in account of customer $1000

Bank A created the $1000 at will out of thin air. This is how it happens most of the time.