Too bad uncle Milty (Milton Friedman) is not still with us, he could answer your question much better than I.
You may have forgotten that the fed has billions of dollars of income every year. Not all existing dollars are in circulation in the U.S. economy. If Treasury borrowing has the net effect of increasing the amount of money in circulation, and it does, inflation can result, and it often does. It depends. For example, just selling bonds to the Fed, via the secondary market, when the Fed uses cash on hand to purchase them (a common transaction) increases the amount of money in circulation. The rapidity of effect then depends, among other things, on the subsequent velocity of the money put into circulation..
QE aside, the Fed routinely buys and sell bonds. When they buy bonds , that increases the amount of money in circulation. When they sell bonds, that decreases the amount of money in circulation.
Although the Fed and Treasury working together may create new money out of thin air as you say to exchange for old money taken out of circulation or to adjust for demographic changes, etc., generally speaking new money is created out of debt, not out of thin air. QE for example creates new money out of debt. Real printing is the creation of money out of thin air. There is a huge difference, but it is seldom appreciated.