I don't have that much of a problem with it, myself. It's not unlike the saying in sports--"You're only as good as your last game," It doesn't matter if you won the championship the past three seasons running--open the next season with a three-game losing streak and the fans--many of whom don't have one-tenth the athletic talent and not one-hundredth the discipline to develop that talent--start calling for your head. It's not fair, and its not right--but it happens.
With superheroes, the stakes are much greater than personal, civic or national pride. There are lives on the line, and if they fail, the city--if not the world--burns. That's what comes with possessing great power and using it for the greater good--people expect you to get it right all the time. And if you don't?
Well, we saw what happened last night, didn't we?
Supergirl has a hard time ahead of her. Apologies won't be enough; explanations won't be enough. She'll have to keep on doing good deeds, and keep on winning. Every time. Eventually, she'll be accepted again--after all, people finally accepted Spider-Man, even though J. Jonah Jameson kept portraying him as a menace. (At least that's how it went in the first three movies--I haev not been following the comics lately.)
That would have been the way to go--except he was already on the scene, and there were too many witnesses. He should have done that at DEO headquarters--preferably away from the prying eyes of that one senator.