Reagan was able to do it, because of how absolutely hated they were in the public conscious because of those previous horrors. It was a double win, an easy target no one would miss AND an appearance of saving tax money.

You can't shut down a popular institution, and those viral stories of mad monster doctors made sure it was an easy sell. Even if they aren't true, rare, or in the past (see anything claimed by Blacks, Women or Gays when they petition for "change" for how this is still used).