Your mistake is to consider the Fed profits instead of the profits derived by the private sector from money lent to the government.
The government is running a deficit, but the law (treaties in Europe) forbid it to fund itself by printing money. So it needs to reduce expenses and/or increase taxes, but in the meantime it has to borrow from the market (whether that happens through the central bank like in the US is irrelevant). Borrowed money is not free at all as you say, it comes with an interest that is paid to the private lender, not to the Fed. The interest adds to the deficit, increasing the drive for higher taxes (hence the burdening of the tax payer) and lower expenses (hence the austerity).
This mode of funding often amounts to a tax on the poor (through spending cuts in social programs for example) and to a lesser extent a tax on the rich (through tax hikes). A government funded directly through printed money is more of a tax on the rich, due to the loss of value of their savings and assets. It also makes it much easier to run an irresponsible budget.
The government is running a deficit, but the law (treaties in Europe) forbid it to fund itself by printing money. So it needs to reduce expenses and/or increase taxes, but in the meantime it has to borrow from the market (whether that happens through the central bank like in the US is irrelevant). Borrowed money is not free at all as you say, it comes with an interest that is paid to the private lender, not to the Fed. The interest adds to the deficit, increasing the drive for higher taxes (hence the burdening of the tax payer) and lower expenses (hence the austerity).
This mode of funding often amounts to a tax on the poor (through spending cuts in social programs for example) and to a lesser extent a tax on the rich (through tax hikes). A government funded directly through printed money is more of a tax on the rich, due to the loss of value of their savings and assets. It also makes it much easier to run an irresponsible budget.