Not true! Take a look at testbp.org and on bp.org.
Every name in the activity stream is using the display name.
eg. https://buddypress.org/community/members/DJPaul/ – Take a look at Paul’s activity stream. What name is displayed in the activity stream?
In fact, some people wanted it the other way around (using usernames instead of display names)!
Do you have the BP Usernames Only plugin installed? If so, disable it.