Fiat currency is simply currency that is not backed by something physical - in itself it has nothing to do with debt.

In modern economies, a central bank can and does create fiat currency out of nothing - literally declaring it into existance. This has nothing to do with debt. And yes it is an advantage for the central bank / government to be able to do this - this advantage is called seigniorage.

Usually the central bank creates money to increase financial liquidity in the banking system, and as such it wants the newly created currency to enter the financial system. The most common way to do that is for the CB to buy something - usually a bond, but it could be anything. The person selling just gets the market value for their product - no special advantage in selling to a central bank vs anyone else. But the total amount of fiat money in the system goes up. In this way the creation of money is often seen to be linked with debt but is not a fundamental link.

Then there is a second, entirely seperate kind of money that is not fiat money which is called bank money. Bank money is entirely based on debt - it is the debt of fiat money. And bank money is the most common sort of money we use, much more common than fiat money e.g. I typically buy larger purchases with bank money (a bank card transferring the IOU of my bank to a shop) instead of with fiat money (notes and coins).

I agree this stuff is entirely not too obvious & the misconception that all money is based on debt is incredibly common.