At bottom it's kind of a kludge, which the US has unfortunately waited until now to test in the courts.
There are a lot of reasons why letting the executive control the Fed would be massively destabilizing. None of those reasons go to the constitutionality of the matter, so courts have generally chosen to sort of handwave it - and the executive hasn't really forced the issue.
Won't be able to depend on those factors much longer...
The US has had a nationally bank since before the constitution was adopted: https://en.wikipedia.org/wiki/First_Bank_of_the_United_State.... The U.S. has had a central bank that controls inflation by performing market operations since 1811: https://en.wikipedia.org/wiki/Second_Bank_of_the_United_Stat....
The core concept of a central bank that influences the economy through its transactions with other banks has a lengthy constitutional pedigree. It predates the 20th century precedent on “independent federal agencies” by 120 years or so.
If some institution predating and then coexisting with the Constitution is enough to grant it a constitutional pedigree, there's a whole lot of other nonsense that would qualify.
More to the point, this doesn't address anything of substance from the discussion above. No one is arguing there shouldn't be a central bank, or that there's no textual/legal support for one.