Meh, divorces happen over words, massive conflicts happen because of it, etc. "Just words" is not a great counterargument for anything.
Saying we don't need safe spaces is basically exactly the same as saying reddit doesn't need moderation. Or a community doesn't need a code of conduct. Sure, you don't need it, but the words used end up forming the cultural base of the group, and I really don't need to go to University 4chan.
That's a remarkably naive way of looking at the history of hate speech and hateful rhetoric both in the United States and globally.
(I'm not saying what happened at Yale approaches hate speech, but the idea that "words can't hurt you, and if they do you're just a child" is, well, pretty child-like in its understanding of social forces.)
Conversely, adults don't need assistance to empathise with other people. Children do.
Adults don't need institutions to protect them from name calling and hurtful words. Children do.