The piece that your missing is that if they do (like what just happened here) it done essentially a loan. And any funds provided will be recovered via a special assessment on banks. Essentially they will increase their insurance rates that banks pay. It is legally required that the banks themselves cover all costs, and FDIC cannot just take money from tax-payers to pay back private banks.
You are mistaken when you say that the taxpayer pays for it or the Fed prints money and pays for the result. That's not what happens at all.
You can read it from the FDIC directly on SVB
> No losses associated with the resolution of Silicon Valley Bank will be borne by taxpayers. Shareholders and certain unsecured debt holders will not be protected. Senior management has also been removed. Any losses to the Deposit Insurance Fund to support uninsured depositors will be recovered by a special assessment on banks, as required by law.