For example I didn't know that Churchill had some pretty horrendous things to say about Indian people until the last couple of years because British cultural attitudes towards him are pretty unflinchingly patriotic. It doesn't diminish that he played a key role in saving us and the rest of the world from the Nazis, but it does balance his image as a man with positive and negative qualities, which is a more realistic image. To tear his statues down says to other people "fuck your heroes, racism is the only dimension I judge people on" - which is much more likely to garner a reactionary response than to initiate dialogue and understanding on the racism of our history, which IMO is the valuable thing you can get out of these conflicts.
Tearing down statues (of people famous for things unrelated to racism) just makes progressives seem incredibly unreasonable, which is already a popular perception, because of exactly things like this.