The unmanned Columbia orbiter would still burn up a few [weeks/months?] later. Of course even an intact unmanned NASA Space Shuttle would burn up... but since

The unmanned Columbia orbiter would still burn up a few [weeks/months?] later. Of course even an intact unmanned NASA Space Shuttle would burn up... but since we are hypothesising irrational reactions the emotional impact of seeing the unmanned orbiter burn up on TV would have likely stemmed much of the criticism. All NASA would have to do is point to the pictures of a burning up unmanned Columbia and say, "With or without people on it, that was going to happen." Particularly after Challenger, I think it would be pretty easy to dismiss any talking heads that criticised the rescue mission as total asshats.