Teams definately do.
When going to Giants games in San Diego, SH straight up says "Tickets will be issued by the San Diego Padres". Complete with the changing prices as you get close to the game that StubHub is about.
Giants sell group tickets all throughout the year, and for a fact, those unsold go onto StubHub a few weeks or so before the game (probably varies depending on popularity of the game). A group of friends that had 25 group tickets, could have had more tix, but I waited too long, and ended up with SH tix in the same row. I would be surprised if other tickets aren't on SH to begin with.
They make their money on those that need ticket confirmation a week or more before the games, and it offsets the losses they make selling in the last 12 hours or so.