That's right. Schools have BROAD authority, and hate speech... ESPECIALLY the sort that might lead to BULLYING, surely can be limited.

I will add:    http://www.firstamendmentschools.org/freedoms/faq.aspx?id=12993 

May schools limit the time, place, and manner of student expression?

Yes, as long as the time, place, and manner regulations are reasonable and nondiscriminatory.

The U.S. Supreme Court has said that "laws regulating the time, place or manner of speech stand on a different footing than laws prohibiting speech altogether."¹First Amendment jurisprudence provides that time, place, and manner restrictions on speech are constitutional if (1) they are content neutral (i.e., they do not treat speech differently based on content); (2) they are narrowly tailored to serve a governmental interest; and (3) they leave open ample alternative means of expression.

Courts will generally grant even more deference to time, place, and manner restrictions in public schools because students do not possess the same level of rights as adults in a public forum. However, the time, place, and manner regulations must still be reasonable. This means that school officials could limit student distribution of material to certain locations and at certain times, but those regulations would need to be both reasonable and nondiscriminatory.

Notes

¹ Linmark Associates, Inc. v. Township of Willingboro, 431 U.S. 85 (1977).