Yes, BUT they do not need to get 1000 USD from anywhere to put 1000 USD in your account. They literally just say you have 1000 USD. You can then electronically transfer that to anyone else's bank. The banks then borrow government money overnight to settle accounts. That's the only place government money comes into it.

Economists actually refer to the USD in your account as bank money because it did not come from the government. You could just as easily think of that money in your account as a bank issued stablecoin pegged to the dollar that's convertible to dollars.