The Fed can't do something like that. It usually takes the involvement of a bankruptcy court, though the FDIC has some powers it can exercise independently. A full Swedish style resolution would almost have certainly required new legislation.
All that said, I agree with your underlying point that just giving insolvent banks money wasn't a great solution at all, though it was minimally sufficient to prevent bank runs at least on formal banks. There were some runs on shadow banking institutions, though in at least one case -- money markets -- the government stopped one by guaranteeing them as though they had been insured banks. That too was a mistake in my opinion.
All that said, I agree with your underlying point that just giving insolvent banks money wasn't a great solution at all, though it was minimally sufficient to prevent bank runs at least on formal banks. There were some runs on shadow banking institutions, though in at least one case -- money markets -- the government stopped one by guaranteeing them as though they had been insured banks. That too was a mistake in my opinion.