Practically no one supports sexual assault. We abhor it, that's why the legal punishment for rape is so severe, why many lynchings of black men were about rape accusations, etc. Sexual assault has thrived in some places - like parts of Hollywood - despite that, not because of it.
Equality for black people was completely different from that: 100 years ago, very few people believed black people were equal or could be equal. It's only recently that the majority position sees things that way, and there are still notable minorities who disagree.
And that's why "changing hearts" isn't the issue here. You already have the hearts of the public, which sees rape as evil. What MeToo achieved was to show that that thing we all hate exists in more places than we thought.
To make this concrete: Weinstein had enough influence in Hollywood and the media to quash stories about his crimes. But once enough women spoke up, the public became aware, and quickly hated Weinstein. The problem was awareness, and the victims fixed that.
MeToo isn't just Harvey Weinstein. He is just one creep who is representative of a larger problem in society.
You can't change the hearts of people easily, although attitudes slowly change over time. When you can't change people's attitudes, the best you can do is make laws, rules, and policies that alleviate some of the ills, until we reach a place where we achieve a more equal society.