Federal Reserve Notes (FRNs, $100 bills) are liabilities of the Fed. You can actually see this on their balance sheet. That Fed liabilities are used as a medium of exchange does not change the essential nature of FRNs as debt.
Most money in the private sector exists as deposits (liabilities of private banks), which are created by loans, which are redeemable for FRNs (or reserves, in a way). Taxes are paid in FRNs, reserves or coins issued by the Treasury. So yes, both public and private debt are used as a medium of exchange. There’s no contradiction here, but there is a connection.
Money’s use is multifaceted. It serves both as a unit of exchange and a means to shift resources from the private to public sector via fiat issuance and taxation. In reality, the ‘unit of exchange’ is mostly a secondary effect of governments issuing currency in exchange for resources while simultaneously imposing a tax to create demand for that currency. The historical record shows that money has always been a function of the authorities or ‘state’ as it existed in various forms. So the raison d’etre is government acquisition of resources and labor.