There's a paper trail. It's just that when the Fed buys something the money is pulled from thin air. When the debt is paid back the principal returns to thin air, the profit goes to the US Treasury.
Not all government debt is purchased by the treasury though – it only does that in special circumstances to moderate the financial system. Most of is is purchased domestically and much of that by other pieces of government.
And you can't keep it secret. It's like supply and demand. More money means prices naturally raise because more people have more money to spend and will be willing to pay a higher price.
Because government debt first has to be bought by someone on the open market, for some market-based price.
Only then can they go to the central bank.
The Central Bank does not buy from the Government.
But there's a systematic problem there I would admit.
So if you no longer have the paper trail of things being bought and sold, who keeps track/makes public the amount of currency in circulation in this case?
If simply printing more money is a bad signal, why not just keep it secret?