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> I don't get it how it was a problem that those companies made much profit. This should have meant they accumulated capital to invest in stuff, so it would have went back to the economy.
The problem was inflation. The Reich didn't want the money going back into circulation as it already had relentlessly growing military spending. Same thing with consumer goods consumption, which is normally a good thing but whose suppression was desirable in those conditions.
> I wonder if fiat money would have been better
They pretty much already had fiat money.
In June 1939, the statutes of the Reichsbank were revised to abolish any formal limitation on the expansion of the money supply. Though the external value of the Reichsmark remained officially at gold parity, the abandonment of the gold standard demanded by Nazi monetary theorists since the 1920s was now finally acknowledged as a reality. Hitler as Fuehrer of the German people was given the power to determine the money supply at will. The path was cleared for unfettered military spending. (p. 239)