I agree with this statement.
But for me, fiat money has utility when it has value. The price of it depends only in economy. Banks can issue more fiat when there is a crisis (expansionary) in order for the households and markets have a PP. They can also hold tight the money making when the economy is reaching its boom called contractionary. So unlike with bitcoin, banks can easily control the circulation of the fiat making its value more/less everyday.
It is to be noted that banks don't issue fiat money. Fiat money is only issued and brought into circulation by government and central bank. However, you are right in saying that banks can control the supply of fiat money in the economy but this is not straight-forward as you said. Banks do control the circulation but using economic tools like interest rate. When economic is under crisis, banks reduce lending rate making it easier for household and markets to raise loans and vice-versa happens when inflation is increasing at pacy rate. But that doesn't affect the value of money, it only affect the PP of money.